A geometric sequence has a common ratio (\(r\)) between consecutive terms.

A geometric series is the sum of the terms in a geometric sequence.

Geometric sequences can be:

• convergent (when \(|r| < 1\), they get smaller and smaller); or
• divergent (when \(r > 1\), they get bigger and bigger); or

Geometric sequences will alternate when \(r < 0\), because they switch between positive and negative terms.

Formulae

The \(n^{th}\) term of a geometric sequence is given by:

\(u_n=ar^{n-1}\)

where \(u_n\) is the \(n^{th}\) term,
\(a\) is the first term, and
\(r\) is the common ratio.

The sum of the first \(n\) terms in a geometric series is given by:

\(S_n=\dfrac{a(1-r^n)}{1-r}, r\neq1\)

where \(S_n\) is the sum of the first \(n\) terms,
\(n\) is the number of terms being added up,
\(a\) is the first term, and
\(r\) is the common ratio.

For a convergent series, the sum to infinity is given by:

\(S_∞=\dfrac{a}{1-r}, |r|<1\)

where \(S_∞\) is the sum to infinity,
\(a\) is the first term, and
\(r\) is the common ratio.

Use of geometric sequences and series in modelling

For example, saving schemes where each year a percentage is paid into an account. e.g. compound interest.