Geometric sequence

Definition

A geometric sequence is a sequence of numbers where the nth term of the sequence is a multiple of the previous term. For example: 1, 2, 4, 8, 16, 32, ... is a geometric sequence because each term is twice the previous term. In this case, 2 is called the common ratio of the sequence.

Summing a Geometric Sequence

The sum of the first $n$ terms of a geometric sequence is given by

$S_n = \frac{a_1(r^{n+1}-1)}{r-1}$

where $a_1$ is the first term in the sequence, and $r$ is the common ratio.

Infinate Geometric Sequences

An infinate geometric sequence is a geometric sequence with an infinate number of terms. These sequences can have sums, sometimes called limits, if $|r|<1$ .