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# Summary: Geometric Sequences

## Key Equations

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

## Key Concepts

• 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 $r$ is given by ${a}_{n}=r{a}_{n - 1}$ for $n\ge 2$ .
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for a geometric sequence with common ratio $r$ is given by ${a}_{n}={a}_{1}{r}^{n - 1}$.
• In application problems, we sometimes alter the explicit formula slightly to ${a}_{n}={a}_{0}{r}^{n}$.

## Glossary

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

## Licenses & Attributions

### CC licensed content, Shared previously

• College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].