# sequence

## Infinite Series

### Sequences and Series

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

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,

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

**Derivation of Formulas**

Let

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