Summary: Geometric Sequences

Key Equations

recursive formula for [latex]nth[/latex] term of a geometric sequence	[latex]{a}_{n}=r{a}_{n - 1},n\ge 2[/latex]
explicit formula for [latex]nth[/latex] term of a geometric sequence	[latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex]

A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
The constant ratio between two consecutive terms is called the common ratio.
The common ratio can be found by dividing any term in the sequence by the previous term.
The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.
A recursive formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[/latex] for [latex]n\ge 2[/latex] .
As with any recursive formula, the initial term of the sequence must be given.
An explicit formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex].
In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[/latex].

common ratio the ratio between any two consecutive terms in a geometric sequence

geometric sequence a sequence in which the ratio of a term to a previous term is a constant

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