# 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] |

## 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 [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].

## 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

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