Key Equations

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