Request a call back

# Sequences and Series

Sequences and Series PDF Notes, Important Questions And Synopsis

## Sequence and Series PDF Notes, Important Questions and Synopsis

SYNOPSIS

1. A sequence is an ordered list of numbers and has the same meaning as in conversational English. A sequence is denoted by <an>(n ≥ 1) = a1,a2,a3, … an.
2. The various numbers occurring in a sequence are called its terms.
3. A sequence containing finite number of terms is called a finite sequence. A finite sequence has a last term.
4. A sequence which is not a finite sequence, i.e. containing infinite number of terms is called an infinite sequence. There is no last term in an infinite sequence.
5. A sequence is said to be an arithmetic progression if every term differs from the preceding term by a constant number. For example, the sequence a1, a2, a3,… an is called an arithmetic sequence or an AP if an+1 = an + d, for all n Î N, where ‘d’ is a constant called the common difference of the AP.
6. ‘A’ is the arithmetic mean of two numbers ‘a’ and ‘b’ if  form an arithmetic progression.
7. A sequence is said to be a geometric progression or GP if the ratio of any of its terms to its preceding term is the same throughout. A constant ratio is the common ratio denoted by ‘r’.
8. If three numbers are in GP, then the middle term is called the geometric mean of the other two.
9. Some Concepts
1. A sequence has a definite first member, second member, third member and so on.
2. The nth term <an> is called the general term of the sequence.
3. Fibonacci sequence 1, 1, 2, 3, 5, 8,… is generated by the recurrence relation given by
a1 = a2 = 1
a3 = a1 + a2…
an = an-2 + an-1, n > 2

4. If the number of terms are three with common difference 'd', then the three terms can be taken as  a – d, a, a + d.

5. If the number of terms are four with common difference '2d', then the terms can be taken as a – 3d, a – d, a + d, a + 3d.

6. If the number of terms are five with common difference 'd', then the terms can be taken as a – 2d, a – d, a, a + d, a + 2d.

7. If the number of terms are six with common difference '2d', then the terms can be taken as a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d.

8. General form of an AP is a, a + d, a + 2d,… a + (n - 1)d, where ‘a’ is called the first term of the AP and ‘d’ is called the common difference of the AP. ‘d’ can be any real number.

9. If d > 0, then the AP is increasing. If d < 0, then the AP is decreasing. If d = 0, then the AP is constant.

10. If ‘a’ is the first term and ‘d’ is the common difference of an AP with 'm' terms, then the nth term from the end is the term from the beginning.

11. General term of a GP is arn-1, where ‘a’ is the first term and r is the common ratio.

12. If the number of terms of a GP is 3 with the common ratio r, then the selection of terms can be .

13. If the number of terms of a GP is 4 with the common ratio r2 , then the selection of terms can be 14. If the number of terms of a GP is 5 with the common ratio r, then the selection of terms can be 10. If a constant is added to each term of an AP, then the resulting sequence is also an AP.
11. If a constant is subtracted from each term of an AP, then the resulting sequence is
also an AP.

12. If each term of an AP is multiplied by a constant, then the resulting sequence is also an AP.

13. If each term of an AP is divided by a non-zero constant, then the resulting sequence is also an AP.

14. The arithmetic mean A of any two numbers ‘a’ and ‘b’ is given by 15. General form of a GP is a, ar, ar2, ar3…, where ‘a’ is the first term and ‘r’ is the constant ratio which can be any non-zero real number.

16. A sequence in geometric progression will remain in geometric progression if each of its terms is multiplied by a non-zero constant.

17. A sequence obtained by multiplying two GPs term by term will result in a GP with a common ratio as the product of the common ratios of the two GPs.

18. Reciprocals of the terms of a given GP form a GP with the common ratio 19. If each term of a GP is raised to the same power, then the resulting sequence also forms a GP.

20. The geometric mean (GM) of any two positive numbers ‘a’ and ‘b’ is given by Some Special Series

1. Sum of the first ‘n’ natural numbers: 2. Sum of the squares of the first n natural numbers: 3. Sum of the cubes of the first n natural numbers: 4. Sum of the powers of 4 of the first n natural numbers: Download complete content for FREE ## JEE Main Video Lectures By Experts

VIEW ALL
JEE Main - Maths
Let Tr be the rth term of an AP for r=1,2,3,.... If for some positive integers m, n. We have Tm=1/n and Tn=1/m, then Tmn equals to Asked by nitinsolanki14102001 | 16 Sep, 2023, 03:01: PM ANSWERED BY EXPERT
JEE Main - Maths
if an= -2/(4n^2)-(16n)+(15),      [(4n^2)-(16n)+(15) dis entire thing is the denominator)] then: opt a) a1+a2+....+a10= 20/51 opt b) a1+a2+.....a10=30/51 opt c) a1+a2+......a25=50/141 opt d) a1+a2+.....a25=60/141 Its more than one ans type q, pls tell me a detailed explanation on how to do it as i din understand
Asked by joanmaria916 | 08 Jul, 2023, 04:37: PM ANSWERED BY EXPERT
JEE Main - Maths
There are (4n + 1) terms in a certain sequence of which the first (2n + 1) terms form an A.P. of common difference 2 and the last (2n + 1) terms are in G.P. of common ratio 1/2 If the middle term of both A.P. and G.P. are the same, then find the mid-term of this sequence.
Asked by vishnuramrs07 | 30 May, 2023, 09:05: PM ANSWERED BY EXPERT
JEE Main - Maths
The sum of the common terms of the following three arithmetic progressions. 3,7,11,15,…………,3993,7,11,15,…………,399 2,5,8,11,.........3592,5,8,11,.........359  and 2,7,12,17,……,1972,7,12,17,……,197, is equal to _____ .
Asked by chiyalaxmi28 | 18 Mar, 2023, 10:34: AM ANSWERED BY EXPERT
JEE Main - Maths
AP the sum of integers from 1 to 100 that are devisible by 2 or 5 is
Asked by thakuranurag0987 | 27 Dec, 2022, 02:31: PM ANSWERED BY EXPERT
JEE Main - Maths
(x-1)/(x+1) + (1/2)(x^2-1)/(x+1)^2 + (1/3)(x^3-1)/(x+1)^3+ ...  infinity Asked by apoorv.bhardwaj100 | 30 Oct, 2022, 08:06: AM ANSWERED BY EXPERT
JEE Main - Maths
. Asked by swayamagarwal2114 | 09 Aug, 2022, 01:44: PM ANSWERED BY EXPERT
JEE Main - Maths
find the next number in the given series. 101,123,147,173,?
Asked by amit7063910121 | 17 Jun, 2022, 03:32: PM ANSWERED BY EXPERT
JEE Main - Maths
find the next number in the given series. 101,123,147,173,?
Asked by amit7063910121 | 17 Jun, 2022, 03:29: PM ANSWERED BY EXPERT
JEE Main - Maths
complete the series 10 , 2 , -6 ,???
Asked by batratanya498 | 04 Jun, 2022, 03:31: PM ANSWERED BY EXPERT