A finite or infinite sequence a1, a2, a3,… with a common ratio r, so that a2/a1=r, a3/a2=r,…. The first term is usually denoted by a. For example, 3, 6, 12, 24, 48,…is the geometric sequence with a=3, r=2. In such a geometric sequence, the n-th term an is given by an=arn−1.
http://www.articlesforeducators.com/dir/mathematics/sequences_series/chess_problem.asp Stories illustrating the dramatic behaviour of geometric sequences.