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

 
(A brief description of the Arithmetic Mean-Geometric Mean Inequality)
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The Arithmetic Mean-Geometric Mean (often abbreviated AM-GM) [[Inequality]] states that the [[Arithmetic Mean]] of a set of positive real numbers is greater than or equal to the [[Geometric Mean]] of the same set of positive real numbers.  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.
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The Arithmetic Mean-Geometric Mean (often abbreviated AM-GM) [[Inequalities | Inequality]] states that the [[Arithmetic Mean]] of a set of positive real numbers is greater than or equal to the [[Geometric Mean]] of the same set of positive real numbers.  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.
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In general, AM-GM states that for a set of positive real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds:
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<math>\displaystyle\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math>
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The AM-GM inequalitiy is a specific case of the [[Power mean inequality]].  It (and the much more general Power Mean Inequality) are 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]].

Revision as of 22:44, 17 June 2006

The Arithmetic Mean-Geometric Mean (often abbreviated AM-GM) Inequality states that the Arithmetic Mean of a set of positive real numbers is greater than or equal to the Geometric Mean of the same set of positive real numbers. For example, for the set $\{9,12,54\}$, the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case.

In general, AM-GM states that for a set of positive real numbers $a_1,a_2,\ldots,a_n$, the following always holds:

$\displaystyle\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}$

The AM-GM inequalitiy is a specific case of the Power mean inequality. It (and the much more general Power Mean Inequality) are used fairly frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.

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