# Geometric sequences

**Geometric sequences**

**Geometric sequences **change from one term to the other by multiplication of a constant amount. This amount is also called the **common ratio**. This common ratio is deduced by dividing any term in the sequence before it.

For example:

8, 12, 18, 27……

The common ratio r in this sequence is worked out by:

*r* = 12 ÷ 8 = 1.5 * *

**The ***n*th term of a geometric sequence

*n*th term of a geometric sequence

In the following example, it is required to find the *n*th term of the following sequence:

3, 6, 12, 24…….

In this sequence, the first term is 3 and moves on to the next sequence by multiplying by 2. This therefore can be written as:

*u*1 = 3

*u2* = 6 = (3 x 2)

*u3* = 12 = (3 x 2 x 2) = 3 x 2²

Therefore this sequence can be written as:

Therefore the *n*th term of a geometric sequence with the first term as *a *common ratio *r* is written as: