# Definition:Geometric Progression

## Definition

The term geometric progression is used to mean one of the following:

### Geometric Sequence

A geometric sequence is a sequence $\sequence {x_n}$ in $\R$ defined as:

$x_n = a r^n$ for $n = 0, 1, 2, 3, \ldots$

Thus its general form is:

$a, ar, ar^2, ar^3, \ldots$

and the general term can be defined recursively as:

$x_n = \begin{cases} a & : n = 0 \\ r x_{n-1} & : n > 0 \\ \end{cases}$

### Geometric Series

Let $\sequence {x_n}$ be a geometric sequence in $\R$:

$x_n = a r^n$ for $n = 0, 1, 2, \ldots$

Then the series defined as:

$\ds \sum_{n \mathop = 0}^\infty x_n = a + a r + a r^2 + \cdots + a r^n + \cdots$

is a geometric series.