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Sequence and Series

Sequence and Series Synopsis

Synopsis

1. Introduction to Sequence
• 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
• The various numbers occurring in a sequence are called its terms.
• The nth term <an> is called the general term of the sequence.
• The expression obtained by adding all the terms of a sequence is called the series associated with the given sequence.

2. Finite and Infinite sequences
• A sequence containing finite number of terms is called a finite sequence. A finite sequence has a last term
• 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.

3. Fibonacci sequence
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. Arithmetic Progression
• A sequence is said to be an arithmetic progression or A.P., if every term differs from the preceding term by a constant number.
• For example, sequence a1, a2, a3,… an is called an arithmetic sequence or an A.P.  If an+1 = an + d, for all n ε N, where ‘d’ is a constant called the common difference of the A.P.

• General term of an A.P.
i. The nth term or general term of an A.P. is an = a + (n – 1)d, where a is the first term and d is the common difference.
ii. General term of an A.P., given its last term ‘λ’ is λ – (n – 1)d.
iii. General form of an A.P. is a, a + d, a + 2d,… a + (n - 1)d. ‘a’ is called the first term of the A.P. and ‘d’ is called the common difference of the A.P. ‘d’ can be any real number.
iv. Let ‘a’ be the first term and ‘d’ be the common difference of an A.P. with 'm' terms. Then the nth term from the end is the (m – n + 1)th term from the beginning.

Sum of n terms of an A.P.
i. Let a, a + d, a + 2d, …, a + (n – 1)d be an A.P.
Then,  where 𝓁 = a + (n – 1) d
ii. A sequence is an A.P. if and only if the sum of its n terms is an expression of the form Xn2 + Yn, where X and Y are constants.

• Arithmetic Mean
i. ‘A’ is the arithmetic mean of two numbers ‘a’ and ‘b’ if a, A and b form an arithmetic progression and it is given by
ii. Let A1, A2, A3, … An be n numbers between a and b such that a, A1, A2, A3, …An, b is an A.P.
These n numbers between a and b are as follows:

Nature of A.P. depending upon the common difference d
i. If d > 0 then the A.P. is increasing.
ii. If d < 0 then the A.P. is decreasing.
iii. If d = 0 then the A.P. is constant.

Selection of Terms
i. If the number of terms are three with common difference ’d’, then the three terms can be taken as a – d, a, a + d.
ii. 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.
iii. If the number of terms are five with common difference 'd', then the terms can be taken as a – 2d, a – d, a + d, a + 2d.
iv. If the number of terms are six with common difference '2d', then the terms can be taken asa – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d.
v. If the terms of an A.P. are selected at regular intervals, then the selected terms form an A.P.
Let an, an+1, an+2 be the consecutive terms of an A.P., then 2an+1 = an + an+2

5.  Properties of Arithmetic Progression
• If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
• If a constant is subtracted from each term of an A.P., then the resulting sequence is also an A.P.
• If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
• If each term of an A.P. is divided by a non–zero constant, then the resulting sequence is also an A.P.

6. Geometric Progression
A sequence is said to be a geometric progression or G.P., if the ratio of any of its term to its preceding term is the same throughout. Constant ratio is common ratio denoted by ‘r’.

General term of a G.P.
i. General form of a G.P. is a, ar, ar2, ar3,... where ‘a’ is the first term and ‘r’ is the constant ratio, ‘r’ can take any non-zero real number.
ii. General term of a G.P. is arn-1 where ‘a’ is the first term and r is the common ratio.
Sum of a G.P.
i. Sum to first n terms of a G.P. is Sn = a + ar + ar² + … + arn-1
♦ If r = 1, Sn = a + a + a + … + a (n terms) = na
♦ If r < 1, Sn
♦ If r > 1, Sn
ii. The sum of an infinite G.P. with first term ‘a’ and common ratio r ( –1 < r < 1) is given by
• Geometric Mean
i. If three numbers are in G.P., then the middle term is called the geometric mean of the other two.
ii. The geometric mean (G.M.) of any two positive numbers ‘a’ and ‘b’ is given by
iii. Let G1, G2,… Gn be n numbers between positive numbers a and b such that a, G1, G2, G3,… Gn, b is a G.P.

Selection of Terms
i. If the number of terms of a G.P. is 3, with the common ratio r, then the selection of terms can be
ii. If the number of terms of a G.P. is 4, with the common ratio  , then the selection of terms can be
iii. If the number of terms of a G.P. is 5, with the common ratio r, then the selection of terms can be

7. Properties of Geometric Progression
A sequence in geometric progression will remain in geometric progression if each of its terms is multiplied by a non-zero constant.
A sequence obtained by multiplying two G.Ps term by term results in a G.P. with the common ratio is equal to the product of the common ratios of the two G.Ps.
The reciprocals of the terms of a given G.P. form a G.P. with common ratio .
If each term of a G.P. be raised to the same power, the resulting sequence also forms a G.P.

8. Relation between A.M. and G.M.
If A and G be the A.M. and G.M. of two given positive real numbers ‘a’ and ‘b’ respectively, then
A ≥ G where .
Hence, the quadratic equation having a and b as its roots is x2 – 2Ax + G2 = 0
Let A and G be A.M. and G.M. of two given positive real numbers ‘a’ and ‘b’, respectively.
Hence, the given numbers are: .
If A.M and G.M of two numbers are in the ratio of m : n, then the given numbers are in the ratio

9. Harmonic Progression
A sequence is said to be a harmonic progression or H.P., if the reciprocal of the numbers form an A.P.
General (nth) term of an H.P.
i. The nth term of an H.P. is the reciprocal of the nth term of the A.P. formed by the reciprocals of the terms of the H.P.
ii. If the given H.P. is  then its nth term is
• Harmonic Mean
i. If a, H, b are three quantities in H.P., then H is said to be the Harmonic Mean between a and b.
ii. a, H, b are in H.P.  are in A.P.
iii. If a1, a2, a3, …, an are n non-zero numbers in H.P., then their harmonic mean will be

10. Relation between A.P., G.P. and H.P.
Three numbers a, b, c will be in A.P., G.P. or H.P. according as
If A, G and H are respectively the Arithmetic, Geometric and Harmonic Means between any two unequal positive numbers, then
i. A, G, H are in G.P.
ii. A > G > H

11. Arithmetico-Geometric Series
A series in which each term is the product of corresponding terms of an A.P. and a G.P. is called an Arithmetico-Geometric series.
General term of an Arithmetico-Geometric series
i. The general or standard form of such a series is
a + (a + d) + (a + 2d)r2 + (a + 3d)r3 + … + {a + (n – 1)d}rn – 1
Here, each term if formed by multiplying the corresponding terms of the two series:
A.P: a + (a + d) + (a + 2d) + … (a + (n – 1)d) + …
G.P: 1 + r + r2 + … + rn – 1 + …
ii. Nth term of an Arithmetico-Geometric series is given by
Tn = {a + (n – 1)d}rn – 1

Sum of an Arithmetico-Geometric series
i. The sum of the n terms of the series is given by

ii. Sum of an infinite Arithmetico-Geometric series given by

12. Special Series
The sum of  first ‘n’ natural numbers is

Sum of squares of the first n natural numbers is

Sum of cubes of first n natural numbers

Sum of powers of 4 of first n natural numbers

Consider the series a1 + a2 + a3 + a4 + ... + an + ...
If the differences a2 - a1, a3 - a2, a4 - a3, ... are in AP, then the nth term is given by
an = an2 + bn + c, where a, b, c are constant.
Consider the series a1 + a2 + a3 + a4 + ... + an + ...
If the differences a2 - a1, a3 - a2, a4 - a3, ... are in G.P., then the nth term is given by
an = arn-1 + bn + c, where a, b, c are constants.
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