# 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 a_{1}=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 a_{1}=4 and r=2 and we have 12 terms.