Difference between revisions of "Geometric mean"
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| − | The | + | 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>. |
| − | The Geometric Mean is a | + | == 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]] | ||
Latest revision as of 22:04, 11 July 2008
The geometric mean of a collection of 
positive real numbers is the th root of the product of the numbers. Note that if is even, we take the positive th root. It is analogous to the arithmetic mean (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers 
and 
is the number 
such that 
, while the geometric mean of the numbers and is the number 
such that 
.
Examples
The geometric mean of the numbers 6, 4, 1 and 2 is ![$\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}$](http://latex.artofproblemsolving.com/f/f/d/ffde0cff2f31a4ba62c966e51e6fd22fe1d905df.png)
.
The geometric mean features prominently in the Arithmetic Mean-Geometric Mean Inequality.
The geometric mean arises in geometry in the following situation: if 
is a chord of circle 
with midpoint 
and divides the diameter passing through it into pieces of length and then the length of line segment 
is the geometric mean of and .
![[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]](http://latex.artofproblemsolving.com/e/0/f/e0f7e268dd5d382995a16399cbc316344ad6f4af.png)
