Definition:Geometric Sequence

Definition
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 & : n > 0 \\ \end{cases}$

Also known as
The usual term is geometric progression, and the abbreviation G.P. is often seen.

However, prefers the term sequence as there is less likelihood of confusing it with geometric series, which the term geometric progression is also often used for.

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