# Geometric mean

The geometric mean of a collection of $n$ positive real numbers is the $n$th root of the product of the numbers. Note that if $n$ is even, we take the positive $n$th root. It is analogous to the arithmetic mean (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers $b$ and $c$ is the number $a$ such that $a + a = b + c$, while the geometric mean of the numbers $b$ and $c$ is the number $g$ such that $g\cdot g = b\cdot c$.

## Examples

The geometric mean of the numbers 6, 4, 1 and 2 is $\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}$.

The geometric mean features prominently in the Arithmetic Mean-Geometric Mean Inequality.

The geometric mean arises in geometry in the following situation: if $AB$ is a chord of circle $O$ with midpoint $M$ and $M$ divides the diameter passing through it into pieces of length $a$ and $b$ then the length of line segment $AM$ is the geometric mean of $a$ and $b$.

$[asy] size(150); pointfontsize=8; pathfontsize=8; pair A=(3,4),B=(3,-4),M=(3,0); D((-5,0)--(5,0)); D(M--B); MC("\sqrt{ab}",D(A--M,orange+linewidth(1)),W); MC("a",D((-5,-0.3)--(3,-0.3),black,Arrows),S); MC("b",D((3,-0.3)--(5,-0.3),black,Arrows),S); D(CR(D((0,0)),5)); D("A",A,N); D("B",B);D("M",M,NE); [/asy]$

The geometric mean also arises in the following common word problem: if a driver travels half the distance of a trip at a speed of $a$ miles per hour and the other half at a speed of $b$ miles per hour, the average speed over the whole trip is the geometric mean of $a$ and $b$. (If the driver spent half the time of the trip at each speed, we would instead get the arithmetic mean.)