Difference between revisions of "Geometric mean"

Latest revision as of 22:04, 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

1966 AHSME Problems/Problem 3

@@ Line 1: / Line 1: @@
-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.
+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 ==
-Find the geometric mean of the numbers <math>x_1, x_2, x_3, x_4 ... x_n</math>
+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>
+The geometric mean features prominently in the [[Arithmetic Mean-Geometric Mean Inequality]].
-Find the geometric mean of the numbers 6, 4, 1 and 2.
+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]].
+<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]]

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Difference between revisions of "Geometric mean"

Latest revision as of 22:04, 11 July 2008

Contents

Examples

Practice Problems

Introductory Problems

See Also