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# Sequence and Series Formula

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. A series is the addition of all the terms of a sequence. Sequence and series are similar to sets but the difference between them is in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms, which can be either finite or infinite. Let us start learning Sequence and series formula.

## Sequence and Series Formula

What are Sequence and Series?

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. The terms of a sequence usually name as aor an, with the subscripted letter i or n being the index. So, the second term of a sequence might be named a2, and a12 would be the twelfth term.

A series termed as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

SN = a1+a2+a3 + .. + an

### Types of Sequence and Series

Sequences: A finite sequence stops at the end of the list of numbers like a1, a2, a3, a4, a5, a6……an. whereas, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..

Series: In a finite series, a finite number of terms are written like a+ a+ a3 + a+ a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a+ a+ a3 + a+ a5 + a6 + ……a+…..

### Some Common Sequences

Arithmetic Sequences:

A sequence in which every term is obtained by adding or subtraction a definite number to the preceding number is an arithmetic sequence.

Geometric Sequences:

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

Harmonic Sequences:

If the reciprocals of all the elements of the sequence form an arithmetic sequence then the series of numbers is said to be in a harmonic sequence.

Fibonacci Numbers:

Fibonacci numbers form a sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

### Sequence and Series Formulas

The sequence of A.P: The nth term an of the Arithmetic Progression (A.P) a, a+d, a+2d,…a, a+d, a+2d,… is given by

an=a+(n–1)d

Where,

 a First-term d Common difference n Position of the term l Last term

Arithmetic Mean: The arithmetic mean between a and b is given by A.M=$$\frac{a+b}{2}$$

The sequence of G.P: The nth term an of the geometric progression a, ar, ar2, ar3,…,  is an=arn–1an=arn–1

The geometric mean between a and b is G.M= ±\sqrt{ab}

Sequence of H.P: The nth term an of the harmonic progression is an= $$\frac{1}{a+(n–1)d}$$

The harmonic mean between a and b is H.M=$$\frac{2ab}{a+b}$$

Series of A.P: If Sn denotes the sum up to n terms of A.P. a, a+d, a+2d,…a, a+d, a+2d,… then

Sn = $$\frac{n}{2}(a+l),$$

Sn = $$\frac{n}{2} [2a+(n–1)d]$$

The sum of n A.M between a and b is  A.M = $$\frac{n(a+b)}{2}$$

Series of G.P: If Sn denotes the sum up to n terms of G.P is Sn=$$\frac{a(1–rn)}{1–r}$$; r≠1 and l=arn

The sum S of infinite geometric series is S=$$\frac{a}{1–r};$$

## Solved Examples

#### Question 1: If 1, 3, 5, 7, 9…… is a sequence, Find Common difference, nth term, 21st term

Solution: Given sequence is, 1, 3, 5, 7, 9……

1. a) common difference d = 3 – 1 = 2
2. b) The nth term of the arithmetic sequence is denoted by the term Tnand is given by Tn = a + (n-1) d,
3. c) 21st term as:  T21= 1 + (21-1)2 = 1+40 = 41.
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