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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''') states that the [[arithmetic mean]] of a non-empty [[set]] of [[nonnegative]] [[real number]]s is greater than or equal to the [[geometric mean]] of the same set.  (Note that in this case the set of numbers is really a [[multiset]], with repetitions of elements allowed.)  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 Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary inequality, generally one of the first ones taught in number theory courses.  
  
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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.
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== Inequality ==
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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:
  
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In general, AM-GM states that for a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds:
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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>
 
<math>\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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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 [[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]].
 
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]].

Revision as of 16:00, 9 October 2007

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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