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