# sequence

## Infinite Series

### Sequences and Series

Sequence is a succession of numbers formed according to some fixed rule. Example is

$1,~ 8,~ 27,~ 64,~ 125,~ ...$

which is a sequence so that the n^{th} term is given by n^{3}.

Series is the indicated sum of a sequence of numbers. Thus,

$a_1 + a_2 + a_3 + ... + a_n + ...$

is the series corresponding to the sequence $a_1,~ a_2,~ a_3,~ ... ,~a_n,~ ...$

**Finite and Infinite Series**

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## Derivation of Sum of Finite and Infinite Geometric Progression

**Geometric Progression, GP**

Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, *r* of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by *r*.

**Eaxamples of GP:**

- 3, 6, 12, 24, … is a geometric progression with
*r*= 2 - 10, -5, 2.5, -1.25, … is a geometric progression with
*r*= -1/2

## Derivation of Sum of Arithmetic Progression

**Arithmetic Progression, AP**

Definition

*d*.

Examples of arithmetic progression are:

- 2, 5, 8, 11,... common difference = 3
- 23, 19, 15, 11,... common difference = -4