# Geometric Mean

Recall that in the proportion

$\frac{a}{b}=\frac{c}{d}$ ,

$b$
and
$c$
are called the
**
means
**
, and
$a$
and
$d$
are called the
**
extremes
**
.

When the means of a proportion are the same number, that number is called the geometric mean of the extremes.

So if

$\frac{p}{x}=\frac{x}{q}$ ,

then cross-multiplying gives ${x}^{2}=pq$ . Taking the square root of both sides, we get $x=\sqrt{pq}$ as the geometric mean of $p$ and $q$ .

More generally, the geometric mean of a set of
$n$
numbers is the
$n$
^{
th
}
root of their product.

**
Example 1:
**

Find the geometric mean of $25$ and $9$

There are two numbers. So, the geometric mean of the two numbers is the
**
square root
**
of their product.

Geometric mean $=\sqrt{25\cdot 9}$

$\begin{array}{l}=\sqrt{225}\\ =15\end{array}$

The geometric mean of $25$ and $9$ is $15$ .

**
Example 2:
**

Find the geometric mean of $4$ , $10$ and $25$ .

There are three numbers. So, the geometric mean of the three numbers is the cube root of their product.

Geometric mean $=\sqrt[3]{4\cdot 10\cdot 25}$

$\begin{array}{l}=\sqrt[3]{1000}\\ =10\end{array}$

The geometric mean of $4$ , $10$ and $25$ is $10$ .