# How do you find two geometric means between 5 and 135?

Nov 25, 2015

We are effectively looking for $a$ and $b$ such that $5$, $a$, $b$, $135$ is a geometric sequence.

This sequence has common ratio $\sqrt[3]{\frac{135}{5}} = 3$, hence $a = 15$ and $b = 45$

#### Explanation:

In a geometric sequence, each intermediate term is the geometric mean of the term before it and the term after it.

So we want to find $a$ and $b$ such that $5$, $a$, $b$, $135$ is a geometric sequence.

If the common ratio is $r$ then:

$a = 5 r$

$b = a r = 5 {r}^{2}$

$135 = b r = 5 {r}^{3}$

Hence ${r}^{3} = \frac{135}{5} = 27$, so $r = \sqrt[3]{27} = 3$

Then $a = 5 r = 15$ and $b = a r = 15 \cdot 3 = 45$