Arithmetic Mean-Geometric Mean Inequality

Revision as of 16:14, 24 July 2006 by MCrawford (talk | contribs) (Altered definition slightly, added equality condition)

The Arithmetic Mean-Geometric Mean Inequality (AM-GM) states that the arithmetic mean of a set of nonnegative 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.

The equality condition of this inequality states that the AM and GM are equal if and only if all members of the set are equal.

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.

See also

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