RD Sharma Solutions for CBSE Class 10 Mathematics chapter 5 - Arithmetic Progressions

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Chapter 5 - Arithmetic Progressions Excercise Ex. 5.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Write the first five terms for the sequence whose nth terms is:

an = n2 - n + 1

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Find the indicated terms in the sequence whose nth terms are :

an = (n - 1) (2 - n) (3 + n); a1, a2, a3

Solution 13

Question 14

Find the indicated terms in the sequence whose nth terms are:

an = (-1)n n; a3, a5, a8

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Chapter 5 - Arithmetic Progressions Excercise Ex. 5.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Justify whether it is true to say that the sequence having following nth term is an A.P.

 

an = 2n - 1

Solution 9

Question 10

Justify whether it is true to say that the sequence having following nth term is an A.P.

 

an = 3n2 + 5

Solution 10

Question 11

Justify whether it is true to say that the sequence having following nth term is an A.P.

 

an = 1 + n + n2

Solution 11

Chapter 5 - Arithmetic Progressions Excercise Ex. 5.3

Question 1

Solution 1



Question 2

Write the arithmetic progression when first term a and common difference d are as follows:

(i) a = 4, d = -3

(ii) a = -1, d = 1/2

(iii) a = -1.5, d = -0.5

Solution 2

Question 3

In which of the following situations, the sequence of numbers formed will form an A.P.?

The cost of digging a well for the first metre is Rs 150 and rises by Rs 20 for each succeeding metre.


Solution 3

Question 4

In which of the following situations, the sequence of numbers formed will form an A.P.?

The amount of air present in the cylinder when a vacuum pump removes each time 1/4 of their remaining in the cylinder.

Solution 4

Question 5

In which of the following situations, the sequence of numbers formed will form an A.P.?

 

Divya deposited Rs.1000 at compound interest at the rate of 10% per annum. The amount at the end of first year, second year, third year, …., and so on.

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Find out whether of the given sequence is an arithemtic progressions. If it is an arithmetic progressions, find out the common difference.

p, p + 90, p + 180, p + 270, ... where p = (999)999

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Chapter 5 - Arithmetic Progressions Excercise Ex. 5.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Which term of the A.P. 4, 9, 14, ... is 254?

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

The 26th, 11th and last term of an A.P. are 0, 3 and  , respectively. Find the common difference and the number of terms.

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Find the 12th term from the end of the following arithmetic progressions:

3, 8, 13, ..., 253

Solution 29

A.P. is 3, 8, 13, ..., 253

We have:

Last term (l) = 253

Common difference (d) = 8 - 3 = 5

Therefore,

12th term from end

       = l - (n - 1)d

       = 253 - (12 - 1) (5)

       = 253 - 55

       = 198

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

The eighth term of an A.P. is half of its second term and the eleventh term exceeds one third of its fourth term by 1. Find the 15th term.

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

The 17th term of an A.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43, find the nth term.

Solution 58

Thus, nth term is given by

an = a + (n - 1)d
an = 3 + (n - 1)4
an = 3 + 4n - 4
an = 4n - 1


Question 59

Find the number of all three digit natural numbers which are divisible by 9.

Solution 59

The smallest three digit number divisible by 9 = 108

The largest three digit number divisible by 9 = 999

 

Here let us write the series in this form,

108, 117, 126, …………….., 999

a = 108, d = 9

 

tn = a + (n - 1)d

999= 108 + (n - 1)9

999 - 108 = (n - 1)9

891 = (n - 1)9

(n - 1) = 99

n = 99 + 1

n = 100

Number of terms divisible by 9

 

Number of all three digit natural numbers divisible by 9 is 100.

Question 60

The 19th term of an A.P. is equal to three times its sixth term. If its 9th term is 19, find the A.P.

Solution 60

 

Question 61

The 9th term of an A.P. is equal to 6 times its second term. If its 5th term is 22, find the A.P.

Solution 61

 

Question 62

The 24th term of an A.P. is twice its 10th term. Show that its 72nd term is 4 times its 15th term.

Solution 62

Let the first term be 'a' and the common difference be 'd'

t24 = a + (24 - 1)d = a + 23d

t10­ = a + (10 - 1)d = a + 9d

t72 = a + (72 - 1)d = a + 71d

t15 = a + (15 - 1)d = a + 14d

 

t24 = 2t10

a + 23d = 2(a + 9d)

a + 23d = 2a + 18d

23d - 18d = 2a - a

5d = a

 

t72 = a + 71d

= 5d + 71d

= 76d

= 20d + 56d

= 4 × 5d + 4 × 14d

= 4(5d + 14d)

= 4(a + 14d)

= 4t15

t72 = 4t15

 

Question 63

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

Solution 63

L.C.M. of 2 and 5 = 10

3- digit number after 100 divisible by 10 = 110

3- digit number before 999 divisible by 10 = 990

 

Let the number of natural numbers be 'n'

990 = 110 + (n - 1)d

990 - 110 = (n - 1) × 10

880 = 10 × (n - 1)

n - 1 = 88

n = 89

 

The number of natural numbers between 110 and 999 which are divisible by 2 and 5 is 89.

Question 64

If the seventh term of an A.P. is 1/9 and its ninth term is 1/7, find its (63)rd term.

Solution 64

Let the first term be 'a' and the common difference be 'd'.

  

  

 

 

 

 

Question 65

The sum of 5th and 9th terms of an A.P. is 30. If its 25th term is three times its 8th term, find the A.P.

Solution 65

Question 66

Find where 0(zero) is a term of the A.P. 40, 37, 34, 31, ...

Solution 66

Let the first term be 'a' and the common difference be 'd'.

a = 40

d = 37 - 40 = - 3

Let the nth term of the series be 0.

tn = a + (n - 1)d

0 = 40 + (n - 1)( - 3)

0 = 40 - 3(n - 1)

3(n - 1) = 40

No term of the series is 0.

Question 67

Find the middle term of the A.P. 213, 205, 197, …, 37.

Solution 67

Given A.P. is 213, 205, 197, …, 37.

Here, first term = a = 213

And, common difference = d = 205 - 213 = -8

an = 37

nth term of an A.P. is given by

a­n = a + (n - 1)d

37 = 213 + (n - 1)(-8)

37 = 213 - 8n + 8

37 = 221 - 8n

8n = 221 - 37

8n = 184

n = 23

So, there are 23 terms in the given A.P.

The middle term is 12th term.

a12 = 213 + (12 - 1)(-8)

= 213 + (11)(-8)

= 213 - 88

= 125

Hence, the middle term is 125.

Question 68

If the 5th term of an A.P. is 31 and 25th term is 140 more than the 5th term, find the A.P.

Solution 68

Let a be the first term and d be the common difference of the A.P.

Then, we have

a5 = 31 and a25 = a5 + 140

a + 4d = 31 and a + 24d = a + 4d + 140

a + 4d = 31 and 20d = 140

a + 4d = 31 and d = 7

a + 4(7) = 31 and d = 7

a + 28 = 31 and d = 7

a = 3 and d = 7

Hence, the A.P. is a, a + d, a + 2d, a + 3d, ……

i.e. 3, 3 + 7, 3 + 2(7), 3 + 3(7), ……

i.e. 3, 10, 17, 24, …..

Question 69

Find the sum of two middle terms of the A.P.

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?

Solution 72

Question 73

Find the 12th term from the end of the A.P. -2, -4, -6, …, -100.

Solution 73

Question 74

For the A.P.: -3, -7, -11, …, can we find a30 - a20 without actually finding a30 and a20? Give reason for your answer.

Solution 74

Question 75

Two A.P.s have the same common difference. The first term of one A.P. is 2 and that of the other is 7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the same as the difference between any two corresponding terms. Why?

Solution 75

Chapter 5 - Arithmetic Progressions Excercise Ex. 5.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Divide 56 in four parts in A.P. such that the ratio of the product of their extremes to the product of their means is 5 : 6.

Solution 8

Question 9

Solution 9

Question 10

Spilt 207 into three parts such that these are in A.P. and the product of the two smaller parts is 4623.

Solution 10

Question 11

The angles of a triangle are in A.P. the greatest angle is twice the least. Find all the angles.

Solution 11

Question 12

The sum of four consecutive numbers in A.P. is 32 and the ratio of the product of the first and last terms to the product of two middle terms is 7 : 15. Find the number.

Solution 12

Chapter 5 - Arithmetic Progressions Excercise Ex. 5.6

Question 1

Find the sum of

The first 15 multiples of 8.

Solution 1

Multiples of 8 are8,16,24,…

Now,

n=15, a=8, d=8

Question 2

Find the sum of

The first 40 positive integers divisible by (a) 3 (b) 5 (c) 6.

Solution 2

a) divisible by 3 3,6,9,… Now, n=40, a=3, d=3

 

b) divisible by 5 5,10,15,… Now, n=40, a=5, d=5

 

c)divisible by 6 6,12,18,… Now, n=40, a=3, d=6

Question 3

Find the sum of

All 3 - digit natural numbers which are divisible by 13.

Solution 3

Three-digit numbers divisible by 13 are 104,117, 130,…988. Now, a=104, l=988

Question 4

Find the sum of

All 3 - digit natural numbers, which are multiples of 11. 

Solution 4

Three-digit numbers which are multiples of 11 are 110,121, 132,…990. Now, a=110, l=990

Question 5

Find the sum of

All 2 - digit natural numbers divisible by 4. 

Solution 5

Two-digit numbers divisible by 4 are 12,16,…96. Now, a=12, l=96

Question 6

Solution 6

Question 7

Let there be an A.P. with first term 'a', common difference 'd'. If an denotes its nth term and Sn the sum of first n terms, find

S22, if d = 22 and a22 = 149 

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Find the sum of last ten terms of the A.P: 8, 10, 12, 14,.., 126.

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Find the sum of the first 15 terms of each of the following sequences having nth term as

yn = 9 - 5n

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

How many terms are there in the A.P. whose first and fifth terms are -14 and 2 respectively and the sum of the terms is 40?

Solution 28

Question 29

How many terms of the A.P. 9, 17, 25, ... must be taken so that their sum is 636?

 

Remark* - Question modified.

Solution 29

Question 30

Solution 30

Question 31

How many terms of the A.P. 27, 24, 21… should be taken so that their sum is zero?

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Find the sum of

(i) the first 15 multiples of 8

(ii) the first 40 positive integers divisible by (a) 3 (b) 5 (c) 6.

(iii) all 3 - digit natural numbers which are divisible by 13.

(iv) all 3 - digit natural numbers, which are multiples of 11.

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Find the sum:

Solution 43

Let 'a' be the first term and 'd' be the common difference.

   

tn = a + (n - 1)d

  

 

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

The first and the last terms of an A.P. are 7 and 49 respectively. If sum of all its terms is 420, find its common difference.

Solution 55

Let the number of terms be 'n', 'a' be the first term and 'd' be the common difference.

tn = a + (n - 1)d

49 = 7 + (n - 1)d

42 = (n - 1)d…..(i)

 

 

840 = n[14 + (n - 1)d]……(ii)

 

Substituting (ii) in (i),

840 = n[14 + 42]

840 = 56n

n = 15

 

Substituting n in (i)

42 = (15 - 1)d

Common difference, d =3

Question 56

The first and the last terms of an A.P. are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.

Solution 56

Let the number of terms be 'n', 'a' be the first term and 'd' be the common difference.

tn = a + (n - 1)d

45 = 5 + (n - 1)d

40 = (n - 1)d…..(i)

 

 

800 = n[10 + (n - 1)d]……(ii)

 

Substituting (ii) in (i),

800 = n[10 + 40]

800 = 50n

n = 16

 

Substituting n in (i)

40 = (16 - 1)d

Question 57

The sum of first q terms of an A.P. is 162. The ratio of its 6th term to its 13th term is 1 : 2. Find the first and 15th term of the A.P.

Solution 57

  

Question 58

If the 10th term of an A.P. is 21 and the sum of its first ten terms is 120, find its nth term.

Solution 58

Let 'a' be the first term and 'd' be the common difference.

tn = a + (n - 1)d

 

t10 = a + (10 - 1)d

21 = a + 9d……(i)

 

120 = 5[2a + 9d]

24 = 2a + 9d………(ii)

 

(ii) - (i)  

a = 3

Substituting a in (i), we get

a + 9d = 21

3 + 9d = 21

9d = 18

d = 2

 

tn = a + (n - 1)d

= 3 + (n - 1)2

= 3 + 2n - 2

tn= 2n + 1

 

Question 59

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P.

Solution 59

Let the first term be 'a' and the common difference be 'd'   

 

63 = 7[a + 3d]

9 = a + 3d……….(i)

 

Sum of the next 7 terms = 161

Sum of the first 14 terms = 63 + 161 = 224

 

224 = 7[2a + 13d]

32 = 2a + 13d………..(ii)

 

Solving (i) and (ii), we get

d = 2, a = 3

 

28th term of the A.P., t28 = a + (28 - 1)d

= 3 + 27 × 2

= 3 + 54

= 57

 

The 28th of the A.P. is 57.

Question 60

The sum of first seven terms of an A.P. is 182. If its 4th and the 17th terms are in the ratio 1: 5, find the A.P.

Solution 60

Let the first term be 'a' and the common difference be 'd'.

 

26 × 2 = [2a + (7 - 1)d]

52 = 2a + 6d

26 = a + 3d……..(i)

 

From (i) and (ii),

13d = 104

d = 8

 

From (i), a = 2

 

The A.P. is 2, 10, 18, 26,…….

Question 61

The nth term of an A.P. is given by (- 4n + 15). Find the sum of the first 20 terms of this A.P.

Solution 61

Let the first term of the A.P. be 'a' and the common difference be 'd'.

 

tn = - 4n + 15

t1 = - 4 × 1 + 15 = 11

t2 = - 4 × 2 + 15 = 7

t3 = - 4 × 3 + 15 = 3

 

Common Difference, d = 7 - 11 = -4

 

 

= 10 × (-54)

= - 540

 

*Note: Answer given in the book is incorrect.

Question 62

Solution 62

Question 63

Sum of the first 14 terms of an A.P. is 1505 and its first term is 10. Find its 25th term.

Solution 63

Question 64

Solution 64

Question 65

Solution 65

Question 66

Find the number of terms of the A.P. - 12, - 9, - 6, …, 21. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

Solution 66

Let the number of the terms be 'n'.

Common Difference, d = - 9 + 12 = 3

 

tn = a + (n - 1)d

21 = - 12 + 3(n - 1)

21 + 12 = 3(n - 1)

3(n - 1) = 33

n - 1 = 11

n = 12

 

Number of terms of the series = 12

 

If 1 is added to each term of the above A.P.,

- 11, - 8, - 5,…….,22

Number of terms in the series, n1 = 12

Sum of all the terms,

The sum of the terms = 66

Question 67

The sum of the first n terms of an A.P. is 3n2 + 6n. Find the nth term of this A.P.

Solution 67

Sum of n terms of the A.P., Sn = 3n2 + 6n

S1 = 3 × 12 + 6 × 1 = 9 = t1 ……(i)

S2 = 3 × 22 + 6 × 2 = 24 = t1 + t2 …….(ii)

S3 = 3 × 32 + 6 × 3 = 45 = t1 + t2 + t3 ……..(iii)

 

From (i), (ii) and (iii),

t1 = 9, t2 = 15, t3 = 21

 

Common difference, d = 15 - 9 = 6

nth of the AP, tn = a + (n - 1)d

= 9 + (n - 1) 6

= 9 + 6n - 6

= 6n + 3

 

Thus, the nth term of the given A.P. = 6n + 3

Question 68

The sum of the first n terms of an A.P. is 5n - n2. Find the nth term of this A.P.

Solution 68

Sn = 5n - n2

S1 = 5 × 1 - 12 = 4 = t1………..(i)

S2 = 5 × 2 - 22 = 6 = t1 +t2………..(ii)

S3 = 5 × 3 - 32 = 6 = t1 + t2 + t……….(iii)

 

From (i), (ii) and (iii),

t1 = 4, t2 = 2, t3 = 0

 

Here a = 4, d = 2 - 4 = - 2

tn = a + (n - 1)d

= 4 + (n - 1)( -2)

= 4 - 2n + 2

= 6 - 2n

Question 69

The sum of the first n terms of an A.P. is 4n2 + 2n. find the nth term of this A.P.

Solution 69

Sn = 4n2 + 2n

S1 = 4 × 12 + 2 × 1 = 6 = t1………….(i)

S2 = 4 × 22 + 2 × 2 = 20 = t1 + t2……….(ii)

S3 = 4 × 32 + 2 × 3 = 42 = t1 + t2 + t3………..(iii)

 

From (i), (ii) and (iii),

t1 = 6, t2 = 14, t3 = 22

 

Here a = 6, d = 14 - 6 = 8

tn = a + (n - 1)d

tn = 6 + (n - 1)8

= 6 + 8n - 8

= 8n - 2

 

*Note: Answer given in the book is incorrect.

 

Question 70

The sum of first n terms of an A.P. is 3n2 + 4n. find the 25th term of this A.P.

Solution 70

Sum of n terms of the A.P., Sn = 3n2 + 4n

S1 = 3 × 12 + 4 × 1 = 7 = t1………(i)

S2 = 3 × 22 + 4 × 2 = 20 = t1 + t…….(ii)

S3 = 3 × 32 + 4 × 3 = 39 = t1 + t2 + t3 …….(iii)

 

From (i), (ii), (iii)

t1 = 7, t2 = 13, t­3 = 19

Common difference, d = 13 - 7 = 6

 

25th of the term of this A.P., t25 = 7 + (25 - 1)6

= 7 + 144 = 151

The 25th term of the A.P. is 151.

Question 71

The sum of first n terms of an A.P. is 5n2 + 3n. If its mth term is 168, find the value of m. Also, find the 20th term of this A.P.

Solution 71

Sum of the terms, Sn = 5n2 + 3n

S1 = 5 × 12 + 3 × 1 = 8 = t1………..(i)

S2 = 5 × 22 + 3 × 2 = 26 = t1 + t2…………..(ii)

S3 = 5 × 32 + 3 × 3 = 54 = t1 + t2 + t3…………(iii)

 

From(i), (ii) and (iii),

t1 = 8, t2 = 18, t3 = 28

Common difference, d = 18 - 8 = 10

 

tm = 168

a + (m - 1)d = 168

8 + (m - 1)×10 = 168

(m - 1) × 10 = 160

m - 1 = 16

m = 17

t20 = a + (20 - 1)d

= 8 + 19 × 10

= 8 + 190

= 198

 

Question 72

The sum of first q terms of an A.P. is 63q - 3q2. If its pth term -60, find the value of p. Also, find the 11th term of this A.P.



Remark* - Question modified.

Solution 72

Let the first term be 'a' and the common difference be 'd'.

Sum of the first 'q' terms, Sq = 63q - 3q2

S1 = 63 × 1 - 3 × 12 = 60 = t1……..(i)

S2 = 63 × 2 - 3 × 22 = 114 = t1 + t2…..(ii)

S3 = 63 × 3 - 3 × 32 = 162 = t1 + t2 + t3 .....(iii)

From (i), (ii) and (iii),

t1 = 60

t2 = 54

t3 = 48

 

Common difference, d = 54 - 60 = - 6

 

tp = a + (p - 1)d

-60 = 60 + (p - 1)( - 6)

- 120 = - 6(p - 1)

p - 1 = 20

p = 21

 

t11 = 60 + (11 - 1)( - 6)

=60 + 10( - 6)

= 60 - 60

= 0

The 11th term of the A.P. is 0.

Question 73

The sum of first m terms of an A.P. is 4m2 - m. If its nth term is 107, find the value of n. Also, find the 21st term of this A.P.

Solution 73

Let the first term of the A.P. be 'a' and the common difference be 'd'

 

Sum of m terms of the A.P., Sm = 4m2 - m

S1 = 4 × 12 - 1 = 3 = t1 …….(i)

S2 = 4 × 22 - 2 = 14 = t1 + t2……..(ii)

S3 = 4 × 32 - 3 = 33 = t1 + t2 + t …….(iii)

 

From (i), (ii) and (iii)

t1 = 3, t2 = 11, t3 = 19

Common difference, d = 11 - 3 = 8

 

tn = 107

a + (n - 1)d = 107

3 + (n - 1)8 = 107

8(n - 1) = 104

n - 1 = 13

n = 14

 

t21 = 3 + (21 - 1)8 = 3 + 160 = 163

Question 74

Solution 74

Question 75

If the sum of first n terms of an A.P. is  then find its nth term. Hence write its 20th term.

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79

Question 80

Solution 80

Question 81

Solution 81

Question 82

Solution 82

Question 83

Solution 83

Question 84

Solution 84

Question 85

Solution 85

Question 86

Find the sum of all integers between 100 and 550 which are not divisible by 9.

Solution 86

Question 87

Find the sum of all integers between 1 and 500 which are multiplies 2 as well as of 5.

Solution 87

Question 88

Find the sum of all integers from 1 to 500 which are multiplies 2 as well as of 5.

Solution 88

Question 89

Find the sum of all integers from 1 to 500 which are multiples of 2 or 5.

Solution 89

 

 

Question 90

Solution 90

Question 91

Solution 91

Question 92

Solution 92

Question 93

Solution 93

Question 94

Solution 94

Question 95

Let there be an A.P. with first term 'a', common difference 'd'. If an denotes its nth term and Sn the sum of first n terms, find

n and an, if a = 2, d = 8 and Sn = 90.

Solution 95

Question 96

Let there be an A.P. with first term 'a', common difference 'd'. If an denotes its nth term and Sn the sum of first n terms, find k, if Sn = 3n2 + 5n and ak = 164.

Solution 96

Question 97

If Sn denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 - S4).

Solution 97

Question 98

A thief, after committing a theft runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5 m/minute every succeeding minute. After how many minutes, the policeman will catch the thief?

Solution 98

Question 99

The sums of first n terms of three A.P.s are S1, S2 and S3. The first term of each is 5 and their common differences are 2, 4 and 6 respectively. Prove that S1 + S3 = 2S2.

Solution 99

Question 100

Resham wanted to save at least Rs. 6500 for sending her daughter to school next year (after 12 months). She saved Rs. 450 in the first month and raised her saying by Rs.20 every next month. How much will she be able to save in next 12 months? Will she be able to send her daughter to the school next year?

Solution 100

Question 101

In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students.

Solution 101

Trees planted by the student in class 1 = 2 + 2 = 4

Trees planted by the student in class 2 = 4 + 4 = 8

Trees planted by the students in class 3 = 6 + 6 = 12

…….

Trees planted by the students in class 12 = 24 + 24 = 48

 

the series will be 4, 8, 12,………., 48

a = 4, Common Difference, d = 8 - 4 = 4

Let 'n' be the number of terms in the series.

48 = 4 + (n - 1)4

44 = 4(n - 1)

n - 1 = 11

n = 12

Sum of the A.P. series,

Number of trees planted by the students = 312

Question 102

Ramkali would need Rs. 1800 for admission fee and books etc., for her daughter to start going to school from the next year. She saved Rs. 50 in the first month of this year and increased her monthly saving by Rs. 20. After a year, how much money will she save? Will she be able to fulfill her dream of sending her daughter to school?

Solution 102

Since, the difference between the savings of two consecutive months is Rs. 20, therefore the series is an A.P.

 

Here, the savings of the first month is Rs. 50

First term, a = 50, Common difference, d = 20

No. of terms = no. of months

No. of terms, n = 12

= 6[100 + 220]

= 6×320

= 1920

 

After a year, Ramakali will save Rs. 1920.

Yes, Ramakali will be able to fulfill her dream of sending her daughter to school.

Question 103

Solution 103

Question 104

Solution 104

Question 105

Solution 105

Question 106

Solution 106

Question 107

Solution 107

Question 108

Solution 108

Question 109

Solution 109

Question 110

If Sn denotes the sum of the first n terms of an A.P., prove that S30 = 3(S20 - S10).

Solution 110

Let the first term of the A.P. be 'a' and the common difference be 'd'.

 

R.H.S.

= 3(S20 - S10)

= 3(10[2a + 19d] - 5[2a + 9d])

= 3(20a + 190d - 10a - 45d)

= 3(10a + 145d)

= 3 × 5(2a + 29d)

= 15[2a + (30 - 1)d]

= S30

= L.H.S.

Question 111

Solve the equation

(-4) + (-1) + 2 + 5 + …. + x = 437.

Solution 111

Question 112

Which term of the A.P. -2, -7, -12,…, will be -77? Find the sum of this A.P. upto the term -77.

Solution 112

Question 113

The sum of the first n terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another A.P. whose first term is -30 and common difference is 8. Find n.

Solution 113

Question 114

The students of a school decided to beautify the school on the annual day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 metre. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?

Solution 114

Chapter 5 - Arithmetic Progressions Excercise 5.57

Question 1

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is 

(a) 87

(b) 88

(c) 89

(d) 90

Solution 1

Let a is first term of an AP and d is common difference.

We know rth term of this AP is = a + (r - 1)d

7th term: a + 6d = 34 (given)           .........(1)

13th term : a + 12d = 64 (given)    .......(2)

Apply (2) - (1)

(a + 12d) - (a + 6d) = 64 - 34

6d = 30

d = 5

from (1)

a + 6(5) = 34

a = 4

18th term = a + (18 - 1)d

              = a + 17d

              = 4 + 17 × 5

              = 89

So, the correct option is (c).

Question 2

If the sum of P terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be

(a) 0

(b) p - q

(c) p + q

(d) - (p + q)

Solution 2

begin mathsize 12px style We space know space that comma space sum space of space straight n space terms space of space an space AP space is space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
where space straight a space is space first space term space and space straight d space is space common space difference.
It space is space given colon
sum space of space straight p space terms space equals space straight q
straight p over 2 open parentheses 2 straight a space plus space open parentheses straight p space minus space 1 close parentheses straight d close parentheses space equals space straight q space space space space space space..... open parentheses 1 close parentheses
and space sum space of space straight q space terms space equals space straight p
straight q over 2 open parentheses 2 straight a space plus space open parentheses straight q space minus space 1 close parentheses straight d close parentheses space equals space straight p space space space space space space space space...... open parentheses 2 close parentheses
To space find colon space sum space of space straight p space plus space straight q space terms
fraction numerator straight p space plus space straight q over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight p space plus space straight q space minus space 1 close parentheses straight d close parentheses space space space space space space space space space........ open parentheses 3 close parentheses
from space left parenthesis 1 right parenthesis
2 straight a space plus space open parentheses straight p space minus space 1 close parentheses straight d space equals space fraction numerator 2 straight q over denominator straight p end fraction space space space space space space space space space........ open parentheses 4 close parentheses
from space open parentheses 2 close parentheses
2 straight a space plus space open parentheses straight q space minus space 1 close parentheses straight d space equals space fraction numerator 2 straight p over denominator straight q end fraction space space space space space space space...... open parentheses 5 close parentheses
Apply space open parentheses 4 close parentheses space minus space open parentheses 5 close parentheses
open parentheses straight p space minus space straight q close parentheses straight d space equals space fraction numerator 2 straight q over denominator straight p end fraction minus fraction numerator 2 straight p over denominator straight q end fraction
open parentheses straight p space minus space straight q close parentheses straight d space equals space 2 open parentheses fraction numerator straight q squared space minus space straight p squared over denominator pq end fraction close parentheses
straight d space equals space fraction numerator negative 2 open parentheses straight p space plus space straight q close parentheses over denominator pq end fraction space space space space space space space space space space space...... open parentheses 6 close parentheses
Putting space this space into space open parentheses 4 close parentheses
2 straight a space plus space open parentheses straight p space minus space 1 close parentheses space fraction numerator open parentheses straight p space plus space straight q close parentheses open parentheses negative 2 close parentheses over denominator pq end fraction equals fraction numerator 2 straight q over denominator straight p end fraction
straight a space equals space fraction numerator open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses over denominator pq end fraction space plus space fraction numerator 2 straight q over denominator straight p end fraction
straight a space equals space fraction numerator open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses space plus space straight q squared over denominator pq end fraction space space space space space space space space space space space space space space space space.... open parentheses 7 close parentheses
from space open parentheses 3 close parentheses comma space open parentheses 6 close parentheses comma space open parentheses 7 close parentheses
fraction numerator straight p space plus space straight q over denominator 2 end fraction open parentheses fraction numerator 2 open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses space plus space 2 straight q squared over denominator pq end fraction space plus space open parentheses straight p space plus space straight q space minus 1 close parentheses fraction numerator open parentheses straight p space plus space straight q close parentheses over denominator pq end fraction open parentheses negative 2 close parentheses close parentheses
equals open parentheses straight p space plus space straight q close parentheses open curly brackets fraction numerator open parentheses straight p space minus space 1 close parentheses space open parentheses straight p space plus space straight q close parentheses space plus space straight q squared space plus space open parentheses negative 1 close parentheses space open parentheses straight p space minus space 1 close parentheses space open parentheses straight p space plus space straight q close parentheses space plus space straight q space open parentheses straight p space plus space straight q close parentheses space open parentheses negative 1 close parentheses over denominator pq end fraction close curly brackets
equals open parentheses straight p space plus space straight q close parentheses space open curly brackets fraction numerator open parentheses straight p space minus space 1 close parentheses space open parentheses straight p space plus space straight q close parentheses space plus space straight q squared space minus space open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses space minus space pq space minus space straight q squared over denominator pq end fraction close curly brackets
equals open parentheses straight p plus space straight q close parentheses space open curly brackets fraction numerator negative pq over denominator pq end fraction close curly brackets
equals negative open parentheses straight p space plus space straight q close parentheses
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 3

If the sum of n terms of an A.P. be 3n+ n and its common difference is 6, then its first term is

(a) 2

(b) 3

(c) 1

(d) 4

Solution 3

begin mathsize 12px style We space know space that comma space for space an space AP space having space first space term space straight a space and space common space difference space straight d. space space
straight S subscript straight n space equals space sum space of space straight n space terms space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
Now comma space according space to space the space question
straight S subscript straight n space equals space 3 straight n squared space plus space straight n space and space straight d space equals space 6
3 straight n squared space plus space straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses 6 close parentheses
rightwards double arrow 3 straight n squared space plus space straight n space equals space na space plus space 3 open parentheses straight n space minus space 1 close parentheses straight n
rightwards double arrow 3 straight n squared space plus space straight n space equals space na space plus space 3 straight n squared space minus space 3 straight n
rightwards double arrow na space equals space 4 straight n
rightwards double arrow straight a space equals space 4
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

 

*Note: Answer given in the book is incorrect.

Question 4

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be

(a) 5

(b) 6

(c) 7

(d) 8

Solution 4

begin mathsize 12px style first space term space straight a space is space 1
last space term space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d space equals space 11
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses straight n space minus space 1 close parentheses straight d space equals space 10
sum space of space straight n space terms space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space equals space 36
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow straight n over 2 open parentheses 2 space plus space 10 close parentheses space equals space 36
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow straight n over 2 open parentheses 12 close parentheses space equals space 36
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space 6 straight n space equals space 36
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight n space equals space 6
So comma space the space correct space option space is space left parenthesis straight b right parenthesis.
end style

Question 5

If the sum of n terms of an A.P. is 3n2 + 5n then which of its terms is 164?

(a) 26th

(b) 27th

(c) 28th

(d) none of these

Solution 5

begin mathsize 12px style sum space of space straight n space terms space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
3 straight n squared space plus space 5 straight n space equals space na space plus space straight n over 2 open parentheses straight n space minus space 1 close parentheses straight d
space space space space space space space space space space space space space space space space space space space space equals space straight d over 2 straight n squared space plus space straight n open parentheses straight a space minus space straight d over 2 close parentheses
on space comparing space the space coefficients space of space straight n squared space and space straight n comma space we space get
straight d over 2 equals 3 space rightwards double arrow box enclose straight d space equals space 6 end enclose space
and space straight a space minus space straight d over 2 space equals space 5 space rightwards double arrow box enclose straight a space equals space 8 end enclose
straight r to the power of th space term space of space an space AP space is space straight a space plus space open parentheses straight r space minus space 1 close parentheses straight d
8 space plus space open parentheses straight r space minus space 1 close parentheses 6 space equals space 164
open parentheses straight r space minus space 1 close parentheses 6 space equals space 156
open parentheses straight r space minus space 1 close parentheses space equals space 26
straight r space equals space 27
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

If the sum of n terms of an A.P. is 2n2 + 5n, then its nth term is

(a) 4n - 3

(b) 3n - 4

(c) 4n + 3

(d) 3n + 4

Solution 6

begin mathsize 12px style sum space of space straight n space terms space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space equals space 2 straight n squared space plus space 5 straight n
straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d over 2 space equals space 2 straight n plus space 5
straight d over 2 straight n space plus space straight a space minus space straight d over 2 space equals space 2 straight n space plus space 5
on space comparison
straight d over 2 equals space 2 space space space space rightwards double arrow space box enclose straight d space equals space 4 end enclose
and space straight a space minus space straight d over 2 space equals space 5
box enclose straight a space equals space 7 end enclose
straight n to the power of th space space end exponent term space of space AP space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 7 space plus space open parentheses straight n space minus space 1 close parentheses 4
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 7 space plus space 4 straight n space minus space 4
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4 straight n space plus space 3
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is

(a) 13

(b) 9

(c) 21

(d) 17

Solution 7

Let the 3 consecutive terms of the AP be a - d, a, a + d.

It is given that

(a - d) + a + (a + d) = 51

3a = 51

a = 17

and (a - d) (a + d) = 273

a- d= 273

172 - d2 = 273

d2 = 289 - 273

d= 16

d = 4

Now 3rd term, a + d is = a + d

                               = 17 + 4

                               = 21

So, the correct option is (c).

Question 8

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

(a) 5, 10, 15, 20

(b) 4, 10, 16, 22

(c) 3, 7, 11, 15

(d) none of these

Solution 8

begin mathsize 12px style Let space 4 space terms space of space AP space be space space straight a space minus space 3 straight d comma space straight a space minus space straight d comma space straight a space plus space straight d comma space straight a space plus space 3 straight d
It space is space given comma
open parentheses straight a space minus space 3 straight d close parentheses space plus space open parentheses straight a space minus space straight d close parentheses space plus space open parentheses straight a space plus space straight d close parentheses space plus space open parentheses straight a space plus space 3 straight d close parentheses space equals space 50
4 straight a space equals space 50
straight a space equals space 25 over 2
Also space open parentheses straight a space plus space 3 straight d close parentheses space equals space 4 open parentheses straight a space minus space 3 straight d close parentheses
straight a space plus space 3 straight d space equals space 4 straight a space minus space 12 straight d
3 straight a space equals space 15 straight d
straight d space equals space straight a over 5
straight d space equals space 5 over 2
Hence space straight a space minus space 3 straight d space equals space 25 over 2 space minus space 15 over 2 space equals space 5
space space space space space space space space space space space space space space space straight a space minus space straight d space equals space 25 over 2 space minus space 5 over 2 space equals space 10
space space space space space space space space space space space space space space space straight a space plus space straight d space equals space 25 over 2 space plus space 5 over 2 space equals space 15
space space space space space space space space space space space space space space straight a space plus space 3 straight d space equals space 25 over 2 space plus space 15 over 2 space equals space 20
4 space terms space are space open parentheses 5 comma space 10 comma space 15 comma space 20 close parentheses
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 9

Let Sdenote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn - kSn-1 + Sn - 2, then k = 

(a) 1

(b) 2

(c) 3

(d) none of these

Solution 9

begin mathsize 12px style straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space space space space space space space space...... open parentheses 1 close parentheses
straight S subscript straight n space minus space 1 end subscript space equals space fraction numerator straight n space minus space 1 over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 2 close parentheses straight d close parentheses space space space space space space......... open parentheses 2 close parentheses
straight S subscript straight n space minus space 2 end subscript space equals space fraction numerator straight n space minus space 2 over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 3 close parentheses straight d close parentheses space space space space space space space........ open parentheses 3 close parentheses
straight d space equals space straight S subscript straight n space minus space straight k space straight S subscript straight n space minus space 1 end subscript space plus space straight S subscript straight n space minus space 2 end subscript
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
straight d space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space minus space straight k fraction numerator open parentheses straight n space minus space 1 close parentheses over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 2 close parentheses straight d close parentheses space plus space fraction numerator straight n space minus space 2 over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 3 close parentheses straight d close parentheses
2 straight d space equals space 2 straight a space open parentheses straight n space minus space straight k open parentheses straight n space minus space 1 close parentheses space plus space straight n space minus space 2 close parentheses plus space straight d space open parentheses straight n open parentheses straight n space minus space 1 close parentheses space minus space straight k open parentheses straight n space minus space 1 close parentheses open parentheses straight n space minus space 2 close parentheses space plus space open parentheses straight n space minus space 2 close parentheses space open parentheses straight n space minus space 3 close parentheses close parentheses
2 straight d space equals space 2 straight a space open parentheses straight n open parentheses 2 space minus space straight k close parentheses space plus space open parentheses straight k space minus space 2 close parentheses close parentheses space plus space straight d open parentheses straight n open parentheses straight n space minus space 1 close parentheses space minus space straight k open parentheses straight n space minus space 1 close parentheses open parentheses straight n space minus space 2 close parentheses space plus space open parentheses straight n space minus space 2 close parentheses open parentheses straight n space minus space 3 close parentheses close parentheses
on space comparison comma space coefficient space of space 2 straight a space must space be space zero
Hence space straight n open parentheses 2 space minus space straight k close parentheses space plus space open parentheses straight k space minus space 2 close parentheses space equals space 0
open parentheses 2 space minus space straight k close parentheses space open parentheses straight n space minus space 1 close parentheses space equals space 0
box enclose straight k space equals space 2 end enclose
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 10

begin mathsize 12px style The space first space and space last space term space of space an space straight A. straight P. space are space straight a space and space straight l space respectively. space If space straight S space is space the space sum space of space all space the space terms space of space the space straight A. straight P. space and space the space common space difference space is space given space by space fraction numerator straight l squared space minus space straight a squared over denominator straight k space minus space open parentheses straight l space plus space straight a close parentheses end fraction comma space then space straight k space equals
open parentheses straight a close parentheses space straight S
open parentheses straight b close parentheses space 2 straight S
open parentheses straight c close parentheses space 3 straight S
open parentheses straight d close parentheses space none space of space these end style

Solution 10

begin mathsize 12px style given space first space term space equals space straight a
last space term space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d space equals space straight l space space where space total space terms space are space straight n space and space common space difference space is space straight d.
rightwards double arrow open parentheses straight n space minus space 1 close parentheses straight d space equals space straight l space minus space straight a space space space space space space space space space space space space space space space....... open parentheses 1 close parentheses
given space straight S space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space space space space space space space space space space....... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses
open parentheses straight n space minus space 1 close parentheses space equals space fraction numerator straight l space minus space straight a over denominator straight d end fraction
straight n space equals space 1 space plus space fraction numerator straight l space minus space straight a over denominator straight d end fraction space space space space space space space space space......... open parentheses 3 close parentheses
from space open parentheses 2 close parentheses space & space open parentheses 3 close parentheses
straight S space equals space open parentheses fraction numerator straight d space plus space straight l space minus space straight a over denominator 2 straight d end fraction close parentheses open parentheses 2 straight a space plus space straight l space minus straight a close parentheses
straight S equals space open parentheses fraction numerator straight d space plus space straight l space minus space straight a over denominator 2 straight d end fraction close parentheses open parentheses straight a space plus space straight l close parentheses
2 dS space equals space straight d open parentheses straight a space plus space straight l close parentheses space plus space straight l squared space minus space straight a squared
straight d open parentheses 2 straight S space minus space open parentheses straight a space plus space straight l close parentheses close parentheses space equals space straight l squared space minus space straight a squared
straight d space equals space fraction numerator straight l squared space minus space straight a squared over denominator 2 straight S space minus space open parentheses straight a space plus space 1 close parentheses end fraction space space space space space space space space space......... open parentheses 4 close parentheses
But space it space is space given space that space straight d space equals space fraction numerator straight l squared space minus space straight a squared over denominator straight k space minus space open parentheses straight l space plus space straight a close parentheses end fraction space space space space space space space space space space space space space...... open parentheses 5 close parentheses
from space open parentheses 4 close parentheses space & space open parentheses 5 close parentheses
box enclose straight K space equals space 2 straight S end enclose
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 11

begin mathsize 12px style If space the space sum space of space first space straight n space even space natural space numbers space is space equal space to space straight k space times space the space sum space of space first space straight n space odd space natural space numbers comma space then space straight k space equals
left parenthesis straight a right parenthesis space 1 over straight n
open parentheses straight b close parentheses space fraction numerator straight n space minus space 1 over denominator straight n end fraction
open parentheses straight c close parentheses space fraction numerator straight n space plus space 1 over denominator 2 straight n end fraction
open parentheses straight d close parentheses space fraction numerator straight n space plus space 1 over denominator straight n end fraction end style

Solution 11

begin mathsize 12px style first space space straight n space even space natural space numbers space are space 2 comma space 4 comma space 6 comma space 8..........
This space is space an space AP space with space first space term space 2 space and space common space difference space 2
so space straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space rightwards double arrow space straight S subscript straight n space equals space straight n over 2 open parentheses 4 space plus space open parentheses straight n space minus space 1 close parentheses 2 close parentheses
straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight n space plus space 2 close parentheses
box enclose straight S subscript straight n space equals space straight n open parentheses straight n space plus space 1 close parentheses end enclose space space space space space space space space space space space space space....... open parentheses 1 close parentheses
first space straight n space odd space natural space numbers space are space 1 comma space 3 comma space 5 comma space 7........
This space is space an space AP space with space first space term space 1 space and space common space difference space 2
straight T subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
space space space space space space space equals space straight n over 2 open parentheses 2 space plus space open parentheses straight n space minus space 1 close parentheses 2 close parentheses
space space space space space space space equals space straight n squared
It space is space given space that space straight S subscript straight n space equals space kT subscript straight n
Hence space straight n open parentheses straight n space plus space 1 close parentheses space equals space kn squared
straight k space equals space fraction numerator straight n space plus space 1 over denominator straight n end fraction
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 12

begin mathsize 12px style If space the space first comma space second space and space last space term space of space an space straight A. straight P. space are space straight a comma space straight b space and space 2 straight a space respectively comma space its space sum space is space
open parentheses straight a close parentheses space fraction numerator ab over denominator 2 open parentheses straight b space minus space straight a close parentheses end fraction
open parentheses straight b close parentheses space fraction numerator ab over denominator straight b space minus space straight a end fraction
open parentheses straight c close parentheses space fraction numerator 3 ab over denominator 2 open parentheses straight b space minus space straight a close parentheses end fraction
open parentheses straight d close parentheses space none space of space these end style

Solution 12

begin mathsize 12px style first space term space equals space straight a
second space term space equals space straight b
last space term space equals space 2 straight a
common space difference space equals space second space term space minus space first space term
box enclose straight d space equals space straight b space minus space straight a end enclose
Last space term space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d
rightwards double arrow straight a space plus space open parentheses straight n space minus space 1 close parentheses space open parentheses straight b space minus space straight a close parentheses space equals space 2 straight a
rightwards double arrow open parentheses straight n space minus space 1 close parentheses space equals space fraction numerator straight a over denominator straight b space minus space straight a end fraction space rightwards double arrow space box enclose straight n space equals space fraction numerator straight b over denominator straight b space minus space straight a end fraction end enclose
sum space of space straight n space terms space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator straight b over denominator 2 open parentheses straight b space minus space straight a close parentheses end fraction open parentheses 2 straight a space plus space open parentheses fraction numerator straight b over denominator straight b space minus space straight a end fraction minus space 1 close parentheses open parentheses straight b space minus space straight a close parentheses close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator straight b over denominator 2 open parentheses straight b space minus space straight a close parentheses end fraction open parentheses 2 straight a space plus space fraction numerator straight a over denominator straight b space minus space straight a end fraction open parentheses straight b space minus space straight a close parentheses close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator straight b over denominator 2 open parentheses straight b space minus space straight a close parentheses end fraction open parentheses 2 straight a space plus space straight a close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 3 ab over denominator 2 open parentheses straight b space minus space straight a close parentheses end fraction
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 5 - Arithmetic Progressions Excercise 5.58

Question 1

begin mathsize 12px style If space straight S subscript 1 space is space the space sum space of space an space arithmetic space progression space of space apostrophe straight n apostrophe space odd space number space of space terms space and space straight S subscript 2 space the space sum space of space the space terms space of space the space series space in space odd space places comma space then space straight S subscript 1 over straight S subscript 2 equals
open parentheses straight a close parentheses space fraction numerator 2 straight n over denominator straight n space plus space 1 end fraction
open parentheses straight b close parentheses space fraction numerator straight n over denominator straight n space plus space 1 end fraction
open parentheses straight c close parentheses space fraction numerator straight n space plus space 1 over denominator 2 straight n end fraction
open parentheses straight d close parentheses space fraction numerator straight n space plus space 1 over denominator straight n end fraction
end style

Solution 1

begin mathsize 12px style Let space straight a space be space first space term space and space straight b space be space common space difference.
straight S subscript 1 space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space space space space space space space space space..... open parentheses 1 close parentheses
straight S subscript 2 space equals space sum space of space terms space of space series space in space odd space places
first space term space equals space straight a
common space difference space equals space 2 straight d
total space number space of space terms space equals space fraction numerator straight n space plus space 1 over denominator 2 end fraction
straight S subscript 2 space equals space fraction numerator straight n space plus space 1 over denominator 4 end fraction open parentheses 2 straight a space plus space open parentheses fraction numerator straight n space plus space 1 over denominator 2 end fraction minus 1 close parentheses 2 straight d close parentheses
straight S subscript 2 space equals space fraction numerator straight n space plus space 1 over denominator 4 end fraction open parentheses 2 straight a space plus space open parentheses fraction numerator straight n space minus space 1 over denominator 2 end fraction close parentheses 2 straight d close parentheses
straight S subscript 2 space equals space fraction numerator straight n space plus space 1 over denominator 4 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space space space space space space space space space space space......... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
straight S subscript 1 over straight S subscript 2 equals space fraction numerator begin display style straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses end style over denominator begin display style fraction numerator straight n space plus space 1 over denominator 4 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses end style end fraction
straight S subscript 1 over straight S subscript 2 space equals space fraction numerator 2 straight n over denominator straight n space plus space 1 end fraction
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

If in an A.P., S= n2p and S= m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to

begin mathsize 12px style open parentheses straight a close parentheses space 1 half straight p cubed end style

(b) mn p

(c) p

(d) (m + n)p2

Solution 2

begin mathsize 12px style straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
straight n squared straight p space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
np space equals space straight a space plus space fraction numerator open parentheses straight n space minus space 1 close parentheses over denominator 2 end fraction straight d space space space space space space space space space...... open parentheses 1 close parentheses
straight S subscript straight m space equals space straight m over 2 open parentheses 2 straight a space plus space open parentheses straight m space minus space 1 close parentheses straight d close parentheses
straight m squared straight p space equals space straight m over 2 open parentheses 2 straight a space plus space open parentheses straight m space minus space 1 close parentheses straight d close parentheses
mp space equals space straight a space plus space fraction numerator open parentheses straight m space minus space 1 close parentheses over denominator 2 end fraction straight d space space space space space space space space space...... open parentheses 2 close parentheses
open parentheses 2 close parentheses space minus space open parentheses 1 close parentheses
open parentheses straight a space plus space fraction numerator open parentheses straight m space minus space 1 close parentheses over denominator 2 end fraction straight d space close parentheses space minus space open parentheses straight a space plus space fraction numerator open parentheses straight n space minus space 1 close parentheses over denominator 2 end fraction straight d space close parentheses space equals space mp space minus space np
straight d open parentheses fraction numerator straight m space minus space 1 over denominator 2 end fraction space minus space fraction numerator straight n space minus space 1 over denominator 2 end fraction close parentheses space equals space open parentheses straight m space minus space straight n close parentheses straight p
straight d open parentheses fraction numerator straight m space minus space straight n over denominator 2 end fraction close parentheses space equals space open parentheses straight m space minus space straight n close parentheses straight p
box enclose straight d space equals space 2 straight p end enclose space space space space space space space space space space space space space space space space space...... open parentheses 3 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 3 close parentheses
np space equals space straight a space plus space fraction numerator open parentheses straight n space minus space 1 close parentheses over denominator 2 end fraction 2 straight p
np space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight p
np space equals space straight a space plus space np space minus space straight p
box enclose straight a space equals space straight p end enclose space space space space space space space space space space space space space..... open parentheses 4 close parentheses
straight S subscript straight p space equals space straight p over 2 open parentheses 2 straight a space plus space open parentheses straight p space minus space 1 close parentheses straight d close parentheses
from space open parentheses 3 close parentheses space & space open parentheses 4 close parentheses
straight S subscript straight p space equals space straight p over 2 open parentheses 2 straight p space plus space open parentheses straight p space minus space 1 close parentheses 2 straight p close parentheses
space space space space space space equals space straight p over 2 open parentheses 2 straight p close parentheses open parentheses straight p close parentheses
space space space space space space equals space straight p cubed
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

If Sdenotes the sum of the first terms of an A.P. If S2n = 3Sn, then S3n : Sis equal to

(a) 4

(b) 6

(c) 8

(d) 10

Solution 3

begin mathsize 12px style straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
straight S subscript 2 straight n end subscript space equals space fraction numerator 2 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 2 straight n space minus space 1 close parentheses straight d close parentheses
straight S subscript 3 straight n end subscript space equals space fraction numerator 3 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses
given space straight S subscript 2 straight n space end subscript equals space 3 straight S subscript straight n
straight n open parentheses 2 straight a space plus space open parentheses 2 straight n space minus space 1 close parentheses straight d close parentheses space equals space fraction numerator 3 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
2 straight a space plus space open parentheses 2 straight n space minus space 1 close parentheses straight d space equals space 3 straight a space plus space 3 over 2 open parentheses straight n space minus space 1 close parentheses straight d
straight d open parentheses 2 straight n minus 1 space fraction numerator negative 3 straight n over denominator 2 end fraction plus space 3 over 2 close parentheses space equals space straight a
straight a space equals space straight d open parentheses straight n over 2 space plus space 1 half close parentheses
straight a space equals space fraction numerator straight d open parentheses straight n space plus space 1 close parentheses over denominator 2 end fraction space space space space space space space space space space space space space....... open parentheses 1 close parentheses
straight S subscript 3 straight n end subscript over straight S subscript straight n space equals space fraction numerator begin display style fraction numerator 3 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses end style over denominator begin display style straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses end style end fraction
space space space space space space space space space space space space equals space fraction numerator 3 open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses over denominator 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d end fraction
from space open parentheses 1 close parentheses
straight S subscript 3 straight n end subscript over straight S subscript straight n equals space fraction numerator 3 open parentheses straight d open parentheses straight n space plus space 1 close parentheses space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses over denominator straight d open parentheses straight n space plus space 1 close parentheses space plus space open parentheses straight n space minus space 1 close parentheses straight d end fraction
space space space space space space space space space space space space equals space fraction numerator 3 straight d open parentheses 4 straight n close parentheses over denominator straight d open parentheses 2 straight n close parentheses end fraction
space space space space space space space space space space space space equals space 6
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 4

In an AP, S= q, S= p and Sdenotes the sum of first r terms. Then, Sp + q is equal to

(a) 0

(b) -(p + q)

(c) p + q

(d) pq

Solution 4

begin mathsize 12px style We space know space comma space sum space of space straight n space terms space of space an space AP space is space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
where space straight a space is space first space term space and space straight d space is space common space difference.
It space is space given colon
sum space of space straight p space terms space equals space straight q
straight p over 2 open parentheses 2 straight a space plus space open parentheses straight p space minus space 1 close parentheses straight d close parentheses space equals space straight q space space space space space space..... open parentheses 1 close parentheses
and space sum space of space straight q space terms space equals space straight p
straight q over 2 open parentheses 2 straight a space plus space open parentheses straight q space minus space 1 close parentheses straight d close parentheses space equals space straight p space space space space space space space space...... open parentheses 2 close parentheses
To space find colon space sum space of space straight p space plus space straight q space terms
fraction numerator straight p space plus space straight q over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight p space plus space straight q space minus space 1 close parentheses straight d close parentheses space space space space space space space space space........ open parentheses 3 close parentheses
from space left parenthesis 1 right parenthesis
2 straight a space plus space open parentheses straight p space minus space 1 close parentheses straight d space equals space fraction numerator 2 straight q over denominator straight p end fraction space space space space space space space space space........ open parentheses 4 close parentheses
from space open parentheses 2 close parentheses
2 straight a space plus space open parentheses straight q space minus space 1 close parentheses straight d space equals space fraction numerator 2 straight p over denominator straight q end fraction space space space space space space space...... open parentheses 5 close parentheses
Apply space open parentheses 4 close parentheses space minus space open parentheses 5 close parentheses
open parentheses straight p space minus space straight q close parentheses straight d space equals space fraction numerator 2 straight q over denominator straight p end fraction minus fraction numerator 2 straight p over denominator straight q end fraction
open parentheses straight p space minus space straight q close parentheses straight d space equals space 2 open parentheses fraction numerator straight q squared space minus space straight p squared over denominator pq end fraction close parentheses
straight d space equals space fraction numerator negative 2 open parentheses straight p space plus space straight q close parentheses over denominator pq end fraction space space space space space space space space space space space...... open parentheses 6 close parentheses
Putting space this space into space open parentheses 4 close parentheses
2 straight a space plus space open parentheses straight p space minus space 1 close parentheses space fraction numerator open parentheses straight p space plus space straight q close parentheses open parentheses negative 2 close parentheses over denominator pq end fraction equals fraction numerator 2 straight q over denominator straight p end fraction
straight a space equals space fraction numerator open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses over denominator pq end fraction space plus space fraction numerator 2 straight q over denominator straight p end fraction
straight a space equals space fraction numerator open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses space plus space straight q squared over denominator pq end fraction space space space space space space space space space space space space space space space space.... open parentheses 7 close parentheses
from space open parentheses 3 close parentheses comma space open parentheses 6 close parentheses comma space open parentheses 7 close parentheses comma space we space have
fraction numerator straight p space plus space straight q over denominator 2 end fraction open parentheses fraction numerator 2 open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses space plus space 2 straight q squared over denominator pq end fraction space plus space open parentheses straight p space plus space straight q space minus 1 close parentheses fraction numerator open parentheses straight p space plus space straight q close parentheses over denominator pq end fraction open parentheses negative 2 close parentheses close parentheses
equals open parentheses straight p space plus space straight q close parentheses open curly brackets fraction numerator open parentheses straight p space minus space 1 close parentheses space open parentheses straight p space plus space straight q close parentheses space plus space straight q squared space plus space open parentheses negative 1 close parentheses space open parentheses straight p space minus space 1 close parentheses space open parentheses straight p space plus space straight q close parentheses space plus space straight q space open parentheses straight p space plus space straight q close parentheses space open parentheses negative 1 close parentheses over denominator pq end fraction close curly brackets
equals open parentheses straight p space plus space straight q close parentheses space open curly brackets fraction numerator open parentheses straight p space minus space 1 close parentheses space open parentheses straight p space plus space straight q close parentheses space plus space straight q squared space minus space open parentheses straight p space minus space 1 close parentheses open parentheses straight p space plus space straight q close parentheses space minus space pq space minus space straight q squared over denominator pq end fraction close curly brackets
equals open parentheses straight p plus space straight q close parentheses space open curly brackets fraction numerator negative pq over denominator pq end fraction close curly brackets
equals negative open parentheses straight p space plus space straight q close parentheses
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

If Sdenotes the sum of the first r terms of an A.P. Then, S3: (S2n - Sn) is

(a) n

(b) 3n

(c) 3

(d) none of these

Solution 5

begin mathsize 12px style straight S subscript 3 straight n end subscript space equals space fraction numerator 3 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses
straight S subscript 2 straight n end subscript space equals space fraction numerator 2 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
straight S subscript 2 straight n end subscript space minus space straight S subscript straight n space equals space fraction numerator 2 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 2 straight n space minus space 1 close parentheses straight d close parentheses space minus space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
space space space space space space space space space space space space space space space space space space space space equals space straight n over 2 open parentheses 4 straight a space plus space 2 open parentheses 2 straight n space minus space 1 close parentheses straight d space minus space 2 straight a space minus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
space space space space space space space space space space space space space space space space space space space space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses 4 straight n space minus space 2 space minus space straight n space plus space 1 close parentheses straight d close parentheses
space space space space space space space space space space space space space space space space space space space space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses space space space space space space space space space space space space space...... open parentheses 1 close parentheses
straight S subscript 3 straight n end subscript space equals space fraction numerator 3 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses
fraction numerator straight S subscript 3 straight n end subscript over denominator straight S subscript 2 straight n end subscript minus space straight S subscript straight n end fraction space equals space fraction numerator begin display style fraction numerator 3 straight n over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses end style over denominator begin display style straight n over 2 open parentheses 2 straight a space plus space open parentheses 3 straight n space minus space 1 close parentheses straight d close parentheses end style end fraction
space space space space space space space space space space space space space space space space space space space space space equals space 3
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 6

If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is

(a) 3200

(b) 1600

(c) 200

(d) 2800

Solution 6

begin mathsize 12px style straight a space equals space 2 space and space straight d space equals space 4 space and space straight n space equals space 40
straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
straight S subscript 40 space equals 40 over 2 open parentheses 2 open parentheses 2 close parentheses space plus space open parentheses 40 space minus space 1 close parentheses 4 close parentheses
space space space space space space space space space space equals space 20 open parentheses 4 space plus space 39 space cross times space 4 close parentheses
space space space space space space space space space space equals space 20 open parentheses 4 space plus space 156 close parentheses
space space space space space space space space space space equals space 20 space cross times space 160
space space space space space space space space space space equals space 3200
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 7

The number of terms of the A.P. 3, 7, 11, 15, .... to be taken so that the sum is 406 is

(a) 5

(b) 10

(c) 12

(d) 14

Solution 7

begin mathsize 12px style For space AP space space space space 3 comma space 7 comma space 11 comma space 15 space......
first space term space straight a space equals space 3
common space difference space straight d space equals space 7 space minus space 3
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4
straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
406 space equals space straight n over 2 open parentheses 6 space plus space open parentheses straight n space minus space 1 close parentheses 4 close parentheses
812 space equals space straight n open parentheses 4 straight n space plus space 2 close parentheses
4 straight n squared space plus space 2 straight n space minus space 812 space equals space 0
2 straight n squared space plus space straight n space minus space 406 space equals space 0
straight n space equals space fraction numerator negative 1 space plus-or-minus space square root of 1 space minus space 4 space cross times space 2 space cross times space open parentheses negative 406 close parentheses end root over denominator 2 left parenthesis 2 right parenthesis end fraction
space space space space space space equals space fraction numerator negative 1 space plus-or-minus space square root of 1 space plus space 3248 end root over denominator 4 end fraction
space space space space space space equals space fraction numerator negative 1 space plus-or-minus space square root of 3249 over denominator 4 end fraction space equals space fraction numerator negative 1 space plus-or-minus space 57 over denominator 4 end fraction
but space straight n space can apostrophe straight t space be space negative space rightwards double arrow straight n space equals space fraction numerator negative 1 space plus space 57 over denominator 4 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight n space equals space 14
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 8

begin mathsize 12px style Sum space of space straight n space terms space of space the space series space square root of 2 space plus space square root of 8 space plus space square root of 18 space plus space square root of 32 space plus space..... space is
open parentheses straight a close parentheses space fraction numerator straight n open parentheses straight n space plus space 1 close parentheses over denominator 2 end fraction
open parentheses straight b close parentheses space 2 straight n open parentheses straight n space plus space 1 close parentheses
open parentheses straight c close parentheses space fraction numerator straight n open parentheses straight n space plus space 1 close parentheses over denominator square root of 2 end fraction
open parentheses straight d close parentheses space 1 end style

Solution 8

begin mathsize 12px style The space series space square root of 2 comma space square root of 8 comma space square root of 18 comma space square root of 32 space........ space space space space can space be space written
square root of 2 comma space 2 square root of 2 comma space 3 square root of 2 comma space 4 square root of 2 space space space space.....
This space is space an space AP comma space with
first space term space open parentheses straight a close parentheses space equals space square root of 2
common space difference space straight d space equals space 2 square root of 2 space minus space square root of 2
straight d space equals space square root of 2
straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
So comma space straight S subscript straight n space equals space straight n over 2 open parentheses 2 square root of 2 space plus space open parentheses straight n space minus space 1 close parentheses square root of 2 close parentheses
straight S subscript straight n space equals space fraction numerator straight n square root of 2 over denominator 2 end fraction open parentheses 2 space plus space open parentheses straight n space minus space 1 close parentheses close parentheses
space space space space space space space equals space fraction numerator straight n open parentheses straight n space plus space 1 close parentheses over denominator square root of 2 end fraction
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 9

The 9th term of an A.P. is 449 and 449th term is 9. The term which is equal to zero is

(a) 501th

(b) 502th

(c) 508th 

(d) none of these

Solution 9

rth term of an AP = a + (r - 1)d

9th term of an AP = a + 8d = 449     ......(1)

449th term of an AP = a + 448d = 9     .......(2)

(2) - (1)

440d = -440

d = -1

from (1)

a = 457

rth term

a + (r - 1)d = 0               (given)

457 + (r - 1)(-1) = 0

457 = r - 1

r = 458

So, the correct option is (d).

*Note: The answer given in the textbook is incorrect.

Question 10

begin mathsize 12px style If space fraction numerator 1 over denominator straight x space plus space 2 end fraction comma space fraction numerator 1 over denominator straight x space plus space 3 end fraction comma space fraction numerator 1 over denominator straight x space plus space 5 end fraction space are space in space straight A. straight P. space Then comma space straight x space equals space
left parenthesis straight a right parenthesis space 5
open parentheses straight b close parentheses space 3
open parentheses straight c close parentheses space 1
open parentheses straight d close parentheses space 2 end style

Solution 10

begin mathsize 12px style Given space terms space are space in space AP comma space so space difference space between space them space are space equal. space Hence
fraction numerator 1 over denominator straight x space plus space 3 end fraction minus fraction numerator 1 over denominator straight x space plus space 2 end fraction space equals space fraction numerator 1 over denominator straight x space plus space 5 end fraction space minus space fraction numerator 1 over denominator straight x space plus space 3 end fraction
fraction numerator 2 over denominator straight x space plus space 3 end fraction space equals space fraction numerator 1 over denominator straight x space plus space 5 end fraction space plus space fraction numerator 1 over denominator straight x space plus space 2 end fraction
fraction numerator 2 over denominator straight x space plus space 3 end fraction space equals space fraction numerator 2 straight x space plus space 7 over denominator open parentheses straight x space plus space 2 close parentheses open parentheses straight x space plus space 5 close parentheses end fraction
2 open parentheses straight x space plus space 2 close parentheses open parentheses straight x space plus space 5 close parentheses space equals space open parentheses 2 straight x space plus space 7 close parentheses open parentheses straight x space plus space 3 close parentheses
2 open parentheses straight x squared space plus space 7 straight x space plus space 10 close parentheses space equals space 2 straight x squared space plus space 13 straight x space plus space 21
2 straight x squared space plus space 14 straight x space plus space 20 space equals space 2 straight x squared space plus space 13 straight x space plus space 21
box enclose straight x space equals space 1 end enclose
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 11

The nth term of an A.P., the sum of whose n terms is Sn, is

(a) S+ Sn - 1

(b) S- Sn - 1

(c) S+ Sn + 1

(d) S- Sn + 1 

Solution 11

begin mathsize 12px style straight n to the power of th space term space equals space straight T subscript straight n space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d space space space space space space space space space space space space space.... open parentheses 1 close parentheses
straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
straight S subscript straight n space minus space 1 end subscript equals space fraction numerator straight n space minus space 1 over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 2 close parentheses straight d close parentheses
straight S subscript straight n minus space straight S subscript straight n space minus 1 end subscript space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space minus space fraction numerator straight n space minus space 1 over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space minus space 2 close parentheses straight d close parentheses
space space space space space space space space space space space space space space space space space space space space space space space equals space na space plus space fraction numerator straight n open parentheses straight n space minus space 1 close parentheses straight d over denominator 2 end fraction space minus space open parentheses straight n space minus space 1 close parentheses space straight a space minus space fraction numerator open parentheses straight n space minus space 1 close parentheses open parentheses straight n space minus space 2 close parentheses straight d over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space space space space space space equals space straight a space plus space fraction numerator open parentheses straight n space minus space 1 close parentheses straight d over denominator 2 end fraction open parentheses straight n space minus space open parentheses straight n space minus space 2 close parentheses close parentheses
space space space space space space space space space space space space space space space space space space space space space space space equals space straight a space plus space fraction numerator open parentheses straight n space minus space 1 close parentheses over denominator 2 end fraction straight d space open parentheses 2 close parentheses
space space space space space space space space space space space space space space space space space space space space space space space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d
On space comparing space with space open parentheses 1 close parentheses
straight T subscript straight n space equals space straight S subscript straight n space minus space straight S subscript straight n space minus space 1 end subscript
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 12

The common difference of an A.P., the sum of whose n terms is Sn, is

(a) S- 2Sn - 1 + Sn - 2

(b) Sn - 2Sn - 1 - Sn - 2

(c) Sn - Sn - 2 

(d) Sn - Sn - 1 

Solution 12

We know, nth term = Sn - Sn - 1        ......(1)

and (n - 1)th  term = Sn - 1 - Sn - 2     ......(2)

We also know that common difference is equal to difference between two consecutive terms

so common difference = difference between nth term and (n - 1)th term

from (1) & (2)

common difference = (S- Sn - 1 ) - (Sn - 1 - Sn - 2)

                            = S- Sn - 1 - Sn - 1 + Sn - 2

                            = S- 2 Sn - 1 + Sn - 2

So, the correct option is (a).

Question 13

begin mathsize 12px style If space the space sums space of space straight n space terms space of space two space arithmetic space progressions space are space in space the space ratio space fraction numerator 3 straight n space plus space 5 over denominator 5 straight n space plus space 7 end fraction comma space then space their space straight n to the power of th space terms space are space in space the space ratio
left parenthesis straight a right parenthesis space fraction numerator 3 straight n space minus space 1 over denominator 5 straight n space minus space 1 end fraction
left parenthesis straight b right parenthesis space fraction numerator 3 straight n space plus space 1 over denominator 5 straight n space plus space 1 end fraction
left parenthesis straight c right parenthesis space fraction numerator 5 straight n space plus space 1 over denominator 3 straight n space plus space 1 end fraction
left parenthesis straight d right parenthesis space fraction numerator 5 straight n space minus space 1 over denominator 3 straight n space minus space 1 end fraction end style

Solution 13

begin mathsize 12px style Let space for space AP 1 space rightwards double arrow space straight a subscript 1 space and space straight d subscript 1 space are space their space first space term space and space common space difference space respectively.
straight S subscript straight n 1 space end subscript equals space straight n over 2 open parentheses 2 straight a subscript 1 space plus space left parenthesis straight n space minus space 1 right parenthesis straight d subscript 1 close parentheses space space space space.... open parentheses 1 close parentheses
and space for space AP 2 space rightwards double arrow straight a subscript 2 space and space straight d subscript 2 space are space their space first space term space and space common space difference space respectively.
straight S subscript straight n 2 end subscript space equals space straight n over 2 open parentheses 2 straight a subscript 2 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 2 close parentheses space space space space space space..... open parentheses 2 close parentheses
Given space straight S subscript straight n 1 end subscript over straight S subscript straight n 2 end subscript space equals space fraction numerator begin display style straight n over 2 end style open parentheses 2 straight a subscript 1 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 1 close parentheses over denominator begin display style straight n over 2 end style open parentheses 2 straight a subscript 2 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 2 close parentheses end fraction equals space fraction numerator 3 straight n space plus space 5 over denominator 5 straight n space plus space 7 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator nd subscript 1 space plus space 2 straight a subscript 1 space minus space straight d subscript 1 over denominator nd subscript 2 space plus space 2 straight a subscript 2 space minus space straight d subscript 2 end fraction space equals space fraction numerator 3 straight n space plus space 5 over denominator 5 straight n space plus space 7 end fraction
On space comparison
rightwards double arrow box enclose straight d subscript 1 space equals space 3 end enclose space and space 2 straight a subscript 1 space minus space straight d subscript 1 space equals space 5
rightwards double arrow space box enclose straight a subscript 1 space end subscript equals space 4 end enclose
and space box enclose straight d subscript 2 space equals space 5 end enclose space and space 2 straight a subscript 2 space minus space straight d subscript 2 space equals space 7
box enclose straight a subscript 2 space equals space 6 end enclose
To space find space colon space straight T subscript straight n 1 end subscript over straight T subscript straight n 2 end subscript equals space fraction numerator straight a subscript 1 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 1 over denominator straight a subscript 2 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 2 end fraction
space space space space space space space space space space space space space space space space space space space straight T subscript straight n 1 end subscript over straight T subscript straight n 2 end subscript space equals space fraction numerator 4 space plus space open parentheses straight n space minus space 1 close parentheses 3 over denominator 6 space plus space open parentheses straight n space minus space 1 close parentheses 5 end fraction
space space space space space space space space space space space space space space space space space space space straight T subscript straight n 1 end subscript over straight T subscript straight n 2 end subscript space equals space fraction numerator 3 straight n space plus space 1 over denominator 5 straight n space plus space 1 end fraction end style

So, the correct option is (b).

Chapter 5 - Arithmetic Progressions Excercise 5.59

Question 1

begin mathsize 12px style If space straight S subscript straight n space denote space the space sum space of space straight n space terms space of space an space straight A. straight P. space with space first space term space straight a space and space common space difference space straight d space such space that space straight S subscript straight x over straight S subscript kx space is space independent space of space straight x comma space then
open parentheses straight a close parentheses space straight d space equals space straight a
open parentheses straight b close parentheses space straight d space equals space 2 straight a
open parentheses straight c close parentheses space straight a space equals space 2 straight d
open parentheses straight d close parentheses space straight d space equals space minus straight a end style

Solution 1

begin mathsize 12px style straight S subscript straight x space equals space straight x over 2 open parentheses 2 straight a space plus space open parentheses straight x space minus space 1 close parentheses straight d close parentheses
straight S subscript kx space equals space kx over 2 open parentheses 2 straight a space plus space open parentheses kx space minus space 1 close parentheses straight d close parentheses
so comma space straight S subscript kx over straight S subscript straight x equals space fraction numerator begin display style kx over 2 open parentheses 2 straight a space plus space open parentheses kx space minus space 1 close parentheses straight d close parentheses end style over denominator begin display style straight x over 2 open parentheses 2 straight a space plus space open parentheses straight x space minus space 1 close parentheses straight d close parentheses end style end fraction
straight S subscript kx over straight S subscript straight x space equals space fraction numerator straight k open parentheses 2 straight a space plus space open parentheses kx space minus space 1 close parentheses straight d close parentheses over denominator 2 straight a space plus space open parentheses straight x space minus space 1 close parentheses straight d end fraction
straight S subscript kx over straight S subscript straight x equals fraction numerator straight k open parentheses 2 straight a space plus space kxd space minus space straight d close parentheses over denominator 2 straight a space plus space xd space minus space straight d end fraction
for space straight S subscript kx over straight S subscript straight x space to space be space independent space of space straight x
straight d space should space cancel space 2 straight a space so space that space straight x space terms space cancel space out
if space straight d space equals space 2 straight a
straight S subscript kx over straight S subscript straight x equals fraction numerator straight k open parentheses kxd close parentheses over denominator xd end fraction
space space space space space space space space space space equals space straight k squared
Yes comma space now space straight S subscript kx over straight S subscript straight x space is space independent space of space straight x space so
straight d space equals space 2 straight a space is space correct space answer.
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 12px style If space the space first space term space of space an space straight A. straight P. space is space straight a space and space straight n to the power of th space term space is space straight b comma space then space its space common space difference space is
left parenthesis straight a right parenthesis space fraction numerator straight b space minus space straight a over denominator straight n space plus space 1 end fraction
open parentheses straight b close parentheses space fraction numerator straight b space minus space straight a over denominator straight n space minus space 1 end fraction
open parentheses straight c close parentheses space fraction numerator straight b space minus space straight a over denominator straight n end fraction
open parentheses straight d close parentheses space fraction numerator straight b space plus space straight a over denominator straight n space minus space 1 end fraction end style

Solution 2

begin mathsize 12px style First space term space equals space straight a
straight n to the power of th space term space equals space straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d space equals space straight b
space space space space space space space space space space space space space space space space space space space space space space space open parentheses straight n space minus space 1 close parentheses straight d space equals space straight b space minus space straight a
space space space space space space space space space space space space space space space space space space space space space space space space straight d space equals space fraction numerator straight b space minus space straight a over denominator straight n space minus space 1 end fraction
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 3

The sum of first n odd natural numbers is

(a) 2n - 1

(b) 2n + 1

(c) n2

(d) n- 1

Solution 3

begin mathsize 12px style First space straight n space odd space natural space numbers space are space 1 comma space 3 comma space 5 comma space 7 comma space 9 space......
first space term space equals space 1
common space difference space equals space 3 space minus space 1
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 2
straight S subscript straight n space equals space straight n over 2 open parentheses 2 open parentheses 1 close parentheses space plus space open parentheses straight n space minus space 1 close parentheses open parentheses 2 close parentheses close parentheses
space space space space space space space equals space straight n over 2 open parentheses 2 space plus space open parentheses straight n space minus space 1 close parentheses 2 close parentheses
space space space space space space space equals space straight n squared over 2 cross times space 2
space space space space space space space equals space straight n squared
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 4

Two A.P.'s have the same common difference. The first term of one of these is 8 and that of the other is 3. The difference between their 30th term is

(a) 11

(b) 3

(c) 8

(d) 5

Solution 4

30th term of first AP whose first term is 8

=8 + (30 - 1)d

=8 + 29d

30th term of second AP whose first term is 3

=3 + (30 - 1)d

=3 + 29d

Difference between these terms

= (8 + 29d) - (3 + 29d)

= 5

So, the correct option is (d).

Question 5

If 18, a, b, -3 are in A.P., the a + b =

(a) 19

(b) 7

(c) 11

(d) 15

Solution 5

If given 4 terms are in AP then difference between consecutive terms must be same.

a - 18 = b - a = -3 - b

a - 18 = b - a              and    b - a = - 3 - b

2a = b + 18    ......(1)  and    2b = a - 3      ......(2)

from (1) & (2)

2(2b + 3) = b + 18

4b + 6 = b + 18

3b = 12

b = 4 and a = 11

so a + b = 15

So, the correct option is (d).

Question 6

begin mathsize 12px style The space sum space of space straight n space terms space of space two space straight A. straight P. apostrophe straight s space are space in space the space ratio space 5 straight n space plus space 9 space colon space 9 straight n space plus space 6. space Then comma space the space ratio space of space their space 18 to the power of th space term space is
left parenthesis straight a right parenthesis space 179 over 321
open parentheses straight b close parentheses space 178 over 321
open parentheses straight c close parentheses space 175 over 321
open parentheses straight d close parentheses space 176 over 321 end style

Solution 6

begin mathsize 12px style Let space straight a subscript 1 space and space straight d subscript 1 space are space first space term space and space common space difference space of space AP 1.
and space straight d subscript 2 space and space straight a subscript 2 space are space common space difference space and space first space term space of space second space AP space respectively.
so space Sn subscript 1 space equals space straight n over 2 open parentheses 2 straight a subscript 1 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 1 close parentheses
and space Sn subscript 2 space equals space straight n over 2 open parentheses 2 straight a subscript 2 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 2 close parentheses
given space Sn subscript 1 over Sn subscript 2 space equals space fraction numerator 5 straight n space plus space 9 over denominator 9 straight n space plus space 6 end fraction
fraction numerator begin display style straight n over 2 end style open parentheses 2 straight a subscript 1 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 1 close parentheses over denominator begin display style straight n over 2 end style open parentheses 2 straight a subscript 2 space plus space open parentheses straight n space minus space 1 close parentheses straight d subscript 2 close parentheses end fraction space equals space fraction numerator 5 straight n space plus space 9 over denominator 9 straight n space plus space 6 end fraction
fraction numerator 2 straight a subscript 1 minus space straight d subscript 1 space plus space nd subscript 1 over denominator 2 straight a subscript 2 space minus space straight d subscript 2 space plus space nd subscript 2 end fraction space equals space fraction numerator 5 straight n space plus space 9 over denominator 9 straight n space plus space 6 end fraction
on space comparison
box enclose straight d subscript 1 space equals space 5 end enclose space space space space & space space 2 straight a subscript 1 space minus space straight d subscript 1 space equals space 9
box enclose straight a subscript 1 equals space 7 end enclose
and space box enclose straight d subscript 2 space equals space 9 end enclose space & space 2 straight a subscript 2 space minus space straight d subscript 2 space equals space 6
box enclose straight a subscript 2 space equals space 15 over 2 end enclose
ratio space of space their space 18 to the power of th space end exponent term
open parentheses straight T subscript 18 close parentheses subscript 1 over open parentheses straight T subscript 18 close parentheses subscript 2 equals space fraction numerator straight a subscript 1 space end subscript plus space open parentheses 18 space minus space 1 close parentheses straight d subscript 1 over denominator straight a subscript 2 space plus space open parentheses 18 space minus space 1 close parentheses straight d subscript 2 end fraction
space space space space space space space space space space space space space space space space space equals space fraction numerator straight a subscript 1 space plus space 17 straight d subscript 1 over denominator straight a subscript 2 space plus space 17 straight d subscript 2 end fraction space
space space space space space space space space space space space space space space space space space equals fraction numerator 7 space plus space 17 space cross times space 5 over denominator begin display style 15 over 2 end style plus space 17 space cross times 9 end fraction
space space space space space space space space space space space space space space space space space equals space fraction numerator 92 over denominator begin display style 321 over 2 end style end fraction equals space 184 over 321 end style

*Note: The correct option is not mentioned in the given choices.

Question 7

begin mathsize 12px style If space fraction numerator 5 space plus space 9 space plus space 13 space plus space.... space to space straight n space terms over denominator 7 space plus 9 space plus space 11 space plus space... space to space open parentheses straight n space plus space 1 close parentheses space terms end fraction equals space 17 over 16 comma space then space straight n space equals
open parentheses straight a close parentheses space 8
open parentheses straight b close parentheses space 7
open parentheses straight c close parentheses space 10
open parentheses straight d close parentheses space 11 end style

Solution 7

begin mathsize 12px style 5 space plus space 9 space plus space 13 space plus space..... space to space straight n space terms space is space an space AP space with space straight a space equals space 5 space and space straight d space equals space 4
So comma space straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses
space space space space space space space space space space space space equals space straight n over 2 open parentheses 10 space plus space open parentheses straight n space minus space 1 close parentheses 4 close parentheses
space space space space space space space space space space space space equals space straight n over 2 open parentheses 4 straight n space plus space 6 close parentheses
space space space space space space space space space space space space equals space straight n open parentheses 2 straight n space plus space 3 close parentheses space space space space space space space space....... open parentheses 1 close parentheses
and space 7 space plus space 9 space plus space 11 space....... space to space open parentheses straight n space plus space 1 close parentheses space terms space is space an space AP space with space straight a space equals space 7 space and space straight d space equals space 2
Hence comma space straight S subscript straight n space plus space 1 end subscript space equals space fraction numerator straight n space plus space 1 over denominator 2 end fraction open parentheses 2 straight a space plus space open parentheses straight n space plus space 1 space minus space 1 close parentheses straight d close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator straight n space plus space 1 over denominator 2 end fraction open parentheses 2 straight a space plus space nd close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator straight n space plus space 1 over denominator 2 end fraction open parentheses 14 space plus space 2 straight n close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses straight n space plus space 1 close parentheses open parentheses straight n space plus space 7 close parentheses space space space space space space space space space space space...... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses space & space from space the space question
fraction numerator straight n open parentheses 2 straight n space plus space 3 close parentheses over denominator open parentheses straight n space plus space 1 close parentheses space open parentheses straight n space plus space 7 close parentheses end fraction space equals space 17 over 16
fraction numerator 2 straight n squared space plus space 3 straight n over denominator straight n squared space plus space 8 straight n space plus space 7 end fraction space equals space 17 over 16
32 straight n squared space plus space 48 straight n space equals space 17 straight n squared space plus space 136 straight n space plus space 119
15 straight n squared space minus space 88 straight n space minus space 119 space equals space 0
15 straight n squared space minus space 105 straight n space plus space 17 straight n space minus space 119 space equals space 0
15 straight n space open parentheses straight n space minus space 7 close parentheses space plus space 17 open parentheses straight n space minus space 7 close parentheses space equals space 0
open parentheses straight n space minus space 7 close parentheses open parentheses 15 straight n space plus space 17 close parentheses space equals space 0
straight n space equals space 7 space or space straight n equals fraction numerator negative 17 over denominator 15 end fraction
but space straight n space can apostrophe straight t space be space negative.
so space box enclose straight n space equals space 7 end enclose
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 8

The sum of n terms of an A.P. is 3n+ 5n, then 164 is its

(a) 24th term

(b) 27th term

(c) 26th term

(d) 25th term

Solution 8

begin mathsize 12px style straight S subscript straight n space equals space straight n over 2 open parentheses 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close parentheses space equals space 3 straight n squared space plus space 5 straight n space space space space space open parentheses given close parentheses
space space space space space space space rightwards double arrow space 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d space equals space 6 straight n space plus space 10
space space space space space space space rightwards double arrow space nd space plus space 2 straight a space minus space straight d space equals space 6 straight n space plus space 10
open parentheses On space comparing space coefficient space of space straight n space and space constant space term close parentheses
box enclose straight d space equals space 6 end enclose space and space 2 straight a space minus space straight d space equals space 10
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space box enclose space space straight a space equals space 8 end enclose
straight r to the power of th space term space equals space straight a space plus space open parentheses straight r space minus space 1 close parentheses straight d
space space space space space space space space space space space space space space space space space space equals space 8 space plus space open parentheses straight r space minus space 1 close parentheses 6
space space space space space space space space space space space space space space space space space space equals space 6 straight r space plus space 2
given space straight r to the power of th space term space equals space 164
so comma space 6 straight r space plus space 2 space equals space 164
space space space space space space space space space space space space space space space space space 6 straight r space equals space 162
space space space space space space space space space space space space space space space space space space space space straight r space equals space 27
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 9

If the nth term of an A.P. is 2n + 1, then the sum of first n terms of the A.P. is

(a) n(n - 2)

(b) n(n + 2)

(c) n(n + 1)

(d) n(n - 1)

Solution 9