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:
- $x_n = \begin{cases} a & : n = 0 \\ r x_{n-1} & : n > 0 \end{cases}$
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.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): geometric progression