# geometric sequence

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}= (a-a^{.}r^{n+1})/(1-r)

When |*r*| < 1, then we also can sum the infinite
series, and it will have the sum *a*/(1-*r*).
(When |*r*| ≥ 1, then the series diverges and
has no sum.)

**See Also:** ArithmeticSequence