Arithmetic Mean-Geometric Mean Inequality

Revision as of 13:34, 18 June 2006 by MCrawford (talk | contribs) (uncapitalised names of means)

The Arithmetic Mean-Geometric Mean (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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