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 number theory courses.

Inequality

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

For a set of nonnegative real numbers $a_1,a_2,\ldots,a_n$, the following always holds:

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

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.

The AM-GM inequalitiy is a specific case of the power mean inequality. Both can be used fairly frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.


See also

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