# Geometric progression

Keywords: Geometric progression

Geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For example: 1, 2, 4,.....,32,64,128,...

$${a}_{1},{a}_{2},{a}_{3},...,{a}_{n-1},{a}_{n},{a}_{n+1},...$$

**n-th term of the sequence:**

$${a}_{n}={a}_{1}\cdot {q}^{n-1}$$

$$\left|{a}_{n}\right|=\sqrt{{a}_{n-1}\cdot {a}_{n+1}},\text{}\text{}n1$$

**The sum of the first n terms:**

$${S}_{n}={a}_{1}\frac{{q}^{n}-1}{q-1},\text{}\text{}q\ne 1$$