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 [[set]] of [[nonnegative]] [[real number]]s is greater than or equal to the [[geometric mean]] of the same set of positive real numbers.  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''') 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.   
  
 
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 [[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 <math>a_1,a_2,\ldots,a_n</math>, the following always holds:
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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:
  
 
<math>\displaystyle\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math>
 
<math>\displaystyle\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)\geq\sqrt[n]{a_1a_2\cdots a_n}</math>
  
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 [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]].
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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 08:39, 25 July 2006

The Arithmetic Mean-Geometric Mean Inequality (AM-GM) states that the arithmetic mean of a non-empty set of nonnegative real numbers 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 $\{9,12,54\}$, 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 nonnegative 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. 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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