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