 ## 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 “ai” or “an“, with the sub-scripted letter “i” or “n” being the “index” or the counter. So the second term of a sequence might be named “a2“, and “a12” would designate the twelfth term.

If the first term of an arithmetic sequence is a1 and the common difference is d, then the nth term of the sequence is given by:

aa(n1)d
An arithmetic series is the sum of an arithmetic sequence. We find the sum by adding the first, a1 and last term, an, divide by 2 in order to get the mean of the two values and then multiply by the number of values, n:
S= (n/2)(a1+an)
If a is the first term and d is the common difference of any arithmetic sequence, then an = a1 + (n – 1)d
And, sum of any arithmetic series is: Sn = (n/2) [2a + (n-1)d]
If r is the common ratio and a is the first term of any geometric sequence, then an = a1rn-1
And, sum of any geometric series is Sn = [a(1-rn)/(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.
1. 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.
2. A sequence Tn is said to be an arithmetic progression (A . P) if there exists a number, say d, such that Tn + 1  Tn = dn = 1. The constant number d is called the common difference of the corresponding A . P
3. 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.
4. If a and d be the first term and common difference of the A . PTn, then the sum of first n terms, Sn is given by Sn = n2 [2a + (n  1)d ], n  N.
5. T1 = S1 and for n > 1, we have Tn = Sn  Sn  1
6. 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.
7. The A.M. between given numbers a and b is equal to (a + b)/2.
8. 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.
9. If a, b, 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. (k1 0)
10. If the sum of three numbers in A.Pis 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 10th term in the series 1, 3, 5, 7, …

Solution:

a = 1
d = 3 – 1 = 2

10th term, t10 = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19

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

Solution:

a = 7
d = 13 – 7 = 6

16th term, t16 = 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, S20

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, a1, a2 … an, b are in AP we can say that a1, a2 … an 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, ar2, ar3, …  [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= [n(n+1)(2n+1)]/6

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