A bgeometric sequence/b has a bcommon ratio/b ((r)) between consecutive terms.

A bgeometric series/b is the bsum/b of the terms in a geometric sequence.

Geometric sequences can be:
[ul][li]bconvergent/b (when (|r| < 1), they get smaller and smaller); or[/li][li]bdivergent/b (when (r > 1), they get bigger and bigger); or[/li]/ul
Geometric sequences will balternate/b when (r < 0), because they switch between positive and negative terms.

[b]uFormulae[/u]/b

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.

[b]uUse of geometric sequences and series in modelling[/u]/b

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

Geometric sequences and series:

(u_n=ar^n-1)

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

(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.

(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.