View source for Geometric mean

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

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

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

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

<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>

== Practice Problems ==
===Introductory Problems===
* [[1966 AHSME Problems/Problem 3]]

== See Also ==
*[[Arithmetic Mean]]
*[[AM-GM]]