Definition:Geometric Progression

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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:

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


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:

$\displaystyle \sum_{n \mathop = 1}^\infty x_n = a + a r + a r^2 + \cdots + a r^n + \cdots$

is a geometric series.


Sources