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:

1. Arithmetic Progression (AP)
2. Geometric Progression (GP)

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

a₁ + a₂ + a₃ + ... + a_(n-1) + a_n

Total Terms: n

First Term: a₁

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 = a₁ + (n - 1)d

S_n = n/2(a₁ + a_n) = n/2[2a₁ + (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 + ar² + ar³ + ar⁴ + ...

a_n = ar^(n-1)