# Use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 8 and the 8th term is 18?

## 12 15 7 5

Apr 4, 2017

If the common ratio is positive, then the $7$th term is $12$.

If not, then it would be $- 12$.

#### Explanation:

In a geometric sequence of positive terms, the middle term of three consecutive terms is the geometric mean of the first and third.

Given two positive numbers $a$ and $b$, their geometric mean is:

$\sqrt{a b}$

In our example we find that the geometric mean of $8$ and $18$ is:

$\sqrt{8 \cdot 18} = \sqrt{144} = 12$

So, assuming the common ratio of the geometric sequence is positive, the $7$th term is $12$.

Note that the $7$th term could also be $- 12$, if the common ratio was $- \frac{3}{2}$ instead of $\frac{3}{2}$.