## Sequence and Series:

In this article, read all you need to know about sequence and series.

## Introduction

An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

2,4,6,8,10….is an arithmetic sequence with the common difference 2.

A sequence may be named or referred to by an upper-case letter such as “A” or “S“. The terms of a sequence are usually named something like “*a _{i}*” or “

*a*“, with the sub-scripted letter “

_{n}*i*” or “

*n*” being the “index” or the counter. So the second term of a sequence might be named “

*a*

_{2}“, and “

*a*

_{12}” would designate the twelfth term.

If the first term of an arithmetic sequence is *a*_{1} and the common difference is *d*, then the *n*th term of the sequence is given by:

_{n }= a

_{1 }+ (n−1)d

_{1}and last term, a

_{n}, divide by 2 in order to get the mean of the two values and then multiply by the number of values, n:

_{n }= (n/2)(a

_{1}+a

_{n})

_{n}= a

_{1}+ (n – 1)d

And, sum of any arithmetic series is: S

_{n}= (n/2) [2a + (n-1)d]

_{n}= a

_{1}r

^{n-1}

_{n}= [a(1-r

^{n})/(1-r)]

**Arithmetic Sequence and Series:**A sequence in which the difference between two consecutive terms remains constant. Eg: 2, 5, 8, 11. 14 is an arithmetic sequence.

- A series in which the difference between any term and its previous term is constant (eg. 2 + 5 + 8 + 11 + 14) is an arithmetic series.
- A sequence T
_{n}is said to be an arithmetic progression (A . P) if there exists a number, say d, such that Tn + 1 − Tn = d, n = 1. The constant number d is called the common difference of the corresponding A . P - If a and d be the first term and common difference of the A . P, then T_n is said to be the last term of the series and the formulafor calculation is Tn = a + (n − 1)d, where n ∈ N.
- If a and d be the first term and common difference of the A . P. Tn, then the sum of first n terms, Sn is given by Sn = n2 [2a + (n − 1)d ], n ∈ N.
- T1 = S1 and for n > 1, we have Tn = Sn − Sn − 1
- If the sequence a, A1, A2, ....., An, b is an A . P., then the numbers A1, A2, ....., An are called the n arithmetic means between a and b.
- The A.M. between given numbers a and b is equal to (a + b)/2.
- The sum of n A.M.s between given numbers a and b is equal to n times the A.M between a and b.
- If a, b, c are in A.P., then for any k, a + k, b + k, c + k are in A.P. a − k, b − k, c − k are in A.P. ka, kb, kc are in A.P. a/k, b/k, c/k are in A.P. (k
^{1}0) - If the sum of three numbers in A.P. is given, then the numbers should be taken as a − d, a, a + d. If the sum of four numbers in A.P. is given, then the numbers should be taken as a − 3d, a − d, a + d, and a + 3d.

## Examples

**Example 1: **Find 10^{th} term in the series 1, 3, 5, 7, …

**Solution: **

a = 1

d = 3 – 1 = 2

10^{th} term, t_{10} = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19

**Example 2: **Find 16^{th} term in the series 7, 13, 19, 25, …

**Solution: **

a = 7

d = 13 – 7 = 6

16^{th} term, t_{16} = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97

**Example 3: **Find the number of terms in the series 8, 12, 16, . . .72

**Solution:**

a = 8

l = 72

d = 12 – 8 = 4

**Example 4: **Find 4 + 7 + 10 + 13 + 16 + . . . up to 20 terms

**Solution: **

a = 4

d = 7 – 4 = 3

Sum of first 20 terms, S_{20}

**Example 5: **Find 6 + 9 + 12 + . . . + 30

**Solution: **

a = 6

l = 30

d = 9 – 6 = 3

n = [(l−a)/d]+1 = [(30−6)/3]+1 = 243 + 1 = 8 + 1 = 9

Sum S

## Arithmetic Mean

If a, b, c are in AP, b is the Arithmetic Mean (AM) between a and c. In this case,

If a, a_{1}, a_{2} … a_{n}, b are in AP we can say that a_{1}, a_{2} … a_{n} are the n Arithmetic Means between a and b.

To solve most of the problems related to AP, the terms can be conveniently taken as

3 terms: (a – d), a, (a +d)

4 terms: (a – 3d), (a – d), (a + d), (a +3d)

5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)

In an AP, sum of terms equidistant from beginning and end will be constant.

## Harmonic Progression

Non-zero numbers are in Harmonic Progression(HP) if are in AP. Harmonic Progression is also known as harmonic sequence.

Three non-zero numbers a, b, c will be in HP, if are in AP

The Harmonic Mean(HM) between two numbers a and b =

If a, b, c are in HP,

## Geometric Progression

Geometric Progression(GP) or Geometric Sequence is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.

A geometric progression(GP) is given by a, ar, ar^{2}, ar^{3}, … [where a = the first term , r = the common ratio]

## Sum of an Infinite Geometric Progression

where a= the first term , r = common ratio

**Example: **Find** **

**Solution: **

a = 1, r =

Here -1 < r < 1. Hence,

## Power Series – Important Formulas

= ∑n^{2 }= [n(n+1)(2n+1)]/6