# Difference between revisions of "Arithmetic Mean-Geometric Mean Inequality"

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− | The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary inequality, generally one of the first ones taught in | + | The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary [[inequality]], generally one of the first ones taught in inequality courses. |

− | == | + | == Theorem == |

− | The AM-GM states that for any | + | The AM-GM states that for any [[set]] of [[positive]] [[real number]]s, the [[arithmetic mean]] of the set is greater than or [[equal]] to the [[geometric mean]] of the set. Algebraically, this is written: |

For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds: | For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds: | ||

<math>\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math> | <math>\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math> | ||

+ | |||

+ | Using the shorthand notation for [[summation]]s and [[product]]s: | ||

+ | |||

+ | <math>\left(\frac{\sum\limits_{i=1}^{n}a_i}{n}\right)\geq\sqrt[n]{\prod\limits_{i=1}^{n}a_i}</math> | ||

For example, for the set <math>\{9,12,54\}</math>, the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case. | For example, for the set <math>\{9,12,54\}</math>, the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case. | ||

− | The [[equality condition]] of this [[inequality]] states that the | + | The [[equality condition]] of this [[inequality]] states that the arithmetic mean and geometric mean are equal [[if and only if]] all members of the set are equal. |

− | + | AMGM can be used fairly frequently to solve [[Olympiad]]-level inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | |

− | == | + | === Proof === |

− | + | {{incomplete|proof}} | |

+ | === Weighted Form === | ||

+ | The weighted form of AMGM is given by using [[weighted average]]s. For example, the weighted arithmetic mean of <math>x</math> and <math>y</math> with <math>3:1</math> is <math>\frac{3y+x}{2\cdot 3}</math> and the geometric is <math>\sqrt[2+3]{xy^3}</math>. AMGM applies to weighted averages. | ||

− | == See | + | ==Extensions== |

+ | *The [[power mean inequality]] is a useful inequality extending on the arithmetic mean side of the inequality. | ||

+ | *The [[root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality]] is an extension on both sides of the inequality with further types of means. | ||

+ | ==Problems== | ||

+ | === Introductory === | ||

+ | === Intermediate === | ||

+ | * Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>. | ||

+ | ([[1983 AIME Problems/Problem 9|Source]]) | ||

+ | === Olympiad === | ||

+ | * Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers. Prove that | ||

+ | <center> | ||

+ | <math> | ||

+ | (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3 | ||

+ | </math>. | ||

+ | </center> | ||

+ | ([[2004 USAMO Problems/Problem 5|Source]]) | ||

+ | == See Also == | ||

* [[RMS-AM-GM-HM]] | * [[RMS-AM-GM-HM]] | ||

* [[Algebra]] | * [[Algebra]] | ||

* [[Inequalities]] | * [[Inequalities]] | ||

+ | |||

+ | ==External Links== | ||

* [http://www.mathideas.org/problems/2006/5/29.pdf Basic Inequalities by Adeel Khan] | * [http://www.mathideas.org/problems/2006/5/29.pdf Basic Inequalities by Adeel Khan] | ||

* [http://www.mathideas.org/problems/2006/5/31.pdf Inequalities: An Application of RMS-AM-GM-HM by Adeel Khan] | * [http://www.mathideas.org/problems/2006/5/31.pdf Inequalities: An Application of RMS-AM-GM-HM by Adeel Khan] | ||

− | |||

− | |||

[[Category:Inequality]] | [[Category:Inequality]] | ||

[[Category:Theorems]] | [[Category:Theorems]] |

## Revision as of 20:53, 20 December 2007

The **Arithmetic Mean-Geometric Mean Inequality** (**AM-GM** or **AMGM**) is an elementary inequality, generally one of the first ones taught in inequality courses.

## Contents

## Theorem

The AM-GM states that for any set of positive real numbers, the arithmetic mean of the set is greater than or equal to the geometric mean of the set. Algebraically, this is written:

For a set of nonnegative real numbers , the following always holds:

Using the shorthand notation for summations and products:

For example, for the set , the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case.

The equality condition of this inequality states that the arithmetic mean and geometric mean are equal if and only if all members of the set are equal.

AMGM can be used fairly frequently to solve Olympiad-level inequality problems, such as those on the USAMO and IMO.

### Proof

### Weighted Form

The weighted form of AMGM is given by using weighted averages. For example, the weighted arithmetic mean of and with is and the geometric is . AMGM applies to weighted averages.

## Extensions

- The power mean inequality is a useful inequality extending on the arithmetic mean side of the inequality.
- The root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality is an extension on both sides of the inequality with further types of means.

## Problems

### Introductory

### Intermediate

- Find the minimum value of for .

(Source)

### Olympiad

- Let , , and be positive real numbers. Prove that

.

(Source)