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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 number theory courses.
#REDIRECT[[AM-GM Inequality]]
== 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 <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>
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 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 [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]].
== See also ==
* [[RMS-AM-GM-HM]]
* [[Algebra]]
* [[Inequalities]]
* [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]
[[Category:Number theory]]

Latest revision as of 16:07, 29 December 2021

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