
Glossary: Prime Pages: Top 5000: 
A geometric sequence is a sequence (finite or infinite) of real numbers for which each term is the previous term multiplied by a constant (called the common ratio). For example, starting with 3 and using a common ratio of 2 we get the finite geometric sequence: 3, 6, 12, 24, 48; and also the infinite sequence
3, 6, 12, 24, 48, ..., 3^{.}2^{n} ...In general, the terms of a geometric sequence have the form a_{n} = a^{.}r^{n} (n=0,1,2,...) for fixed numbers a and r. When we add the terms of a geometric sequence, we get a geometric series. If it is a finite series, then we add its terms to get the series' sum a + a^{.}r + a^{.}r^{2} + ... + a^{.}r^{n} = (aa^{.}r^{n+1})/(1r)When r < 1, then we also can sum the infinite series, and it will have the sum a/(1r). (When r > 1, then the series diverges and has no sum.)
See Also: ArithmeticSequence
Chris K. Caldwell © 19992018 (all rights reserved)
