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 $x^{2} = p q$ . Taking the square root of both sides, we get $x = \sqrt{p q}$ 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 $10$ .