# Progressions

A Progression is a sequence of numbers which have some kind of relation. This relation determines what kind of a progression is.

Generally, there are two types of progressions:

- Arithmetic Progression (AP)
- Geometric Progression (GP)

Any progression (AP or GP) can be generally expressed as

a_{1} + a_{2} + a_{3} + ... + a_{(n-1)} + a_{n}

Total Terms: n

First Term: a_{1}

Last Term: a_{n}

### Arithmetic Progression

In AP, the relation among sequence of numbers is that the difference between any two successive numbers is same.

Example: 3, 5, 7, 9, 11, 13, ... is an AP with difference 2. This difference is called common difference.

a_{n} = a_{1} + (n - 1)d

S_{n} = n/2(a_{1} + a_{n}) = n/2[2a_{1} + (n-1)d]

### Geometric Progression

In GP, each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio.

Example: 2, 6, 18, 54, ...

a + ar + ar^{2} + ar^{3} + ar^{4} + ...

a_{n} = ar^{(n-1)}