Difference between revisions of "Geometric mean"

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Given a set of ''n'' numbers, the '''Geometric Mean''' is the ''nth'' root of the product of the numbers. It is analogous to the [[Arithmetic Mean]], except with products.  
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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>.
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== Examples ==
 
== Examples ==
Find the geometric mean of the numbers <math>x_1, x_2, x_3, x_4 ... x_n</math>
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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>.
We want the nth root of the product of the n numbers. There are n numbers so our geometric mean would be <math>\sqrt[n]{x_1 x_2 x_3 x_4 ... x_n}</math>
 
 
 
  
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The geometric mean features prominently in the [[Arithmetic Mean-Geometric Mean Inequality]].
  
Find the geometric mean of the numbers 6, 4, 1 and 2.
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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>.
There are 4 numbers, so we want the 4th root. The numbers' product is 48, so our answer is <math>\sqrt[4]{48}=2\sqrt[4]{3}</math>
 
  
The Geometric Mean is a component of the well-known [[Arithmetic Mean-Geometric Mean]] [[Inequality]].
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<asy>
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size(150);
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pointfontsize=8;
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pathfontsize=8;
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pair A=(3,4),B=(3,-4),M=(3,0);
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D((-5,0)--(5,0)); D(M--B);
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MC("\sqrt{ab}",D(A--M,orange+linewidth(1)),W);
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MC("a",D((-5,-0.3)--(3,-0.3),black,Arrows),S);
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MC("b",D((3,-0.3)--(5,-0.3),black,Arrows),S);
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D(CR(D((0,0)),5));
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D("A",A,N); D("B",B);D("M",M,NE);
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</asy>
  
 
== Practice Problems ==
 
== Practice Problems ==
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===Introductory Problems===
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* [[1966 AHSME Problems/Problem 3]]
  
 
== See Also ==
 
== See Also ==
 
*[[Arithmetic Mean]]
 
*[[Arithmetic Mean]]
 
*[[AM-GM]]
 
*[[AM-GM]]

Latest revision as of 21:04, 11 July 2008

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]

Practice Problems

Introductory Problems

See Also

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