# Geometric sequences and series

A geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio, r.

$a_{n}=a_{n-1}\cdot r\;\: or\;\: a_{n}=a_{1}\cdot r^{n-1}$

Example

Write the first five terms of a geometric sequence in which a1=2 and r=3.

We use the first given formula:

$a_{1}=2$

$a_{2}=2\cdot 3=6$

$a_{3}=6\cdot 3=18$

$a_{4}=18\cdot 3=54$

$a_{5}=54\cdot 3=162$

Just as with arithmetic series it is possible to find the sum of a geometric series. It is found by using one of the following formulas:

$S_{n}=\frac{a_{1}-a_{1}\cdot r^{n}}{1-r}\; \; or\; \; S_{n}=\frac{a_{1}(1-r^{n})}{1-r}$

## Video lesson

Use the formula for the sum of a geometric series to determine the sum when a1=4 and r=2 and we have 12 terms.