Definition:Geometric Sequence

Definition
A geometric progression is a sequence $\left \langle{x_n}\right \rangle$ in $\R$ defined as:
 * $x_n = a r^{n-1}$ for $n = 1, 2, 3, \ldots$

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 = 1 \\ r x_{n - 1} & : n > 1 \\ \end{cases}$

Also known as
Euclid used the term continued proportion throughout Book $\text{VIII}$ of, though never formally defining it.