During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.

Difference between revisions of "Arithmetic Mean-Geometric Mean Inequality"

(Redirected to a page with a prettier name)
(Tag: New redirect)
 
(27 intermediate revisions by 15 users not shown)
Line 1: Line 1:
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]
 
 
 
{{wikify}}
 
 
 
{{stub}}
 
[[Category:Number theory]]
 
[[Category:Theorems]]
 
[[Category:Algebra]]
 
[[Category:Inequality]]
 

Latest revision as of 16:07, 29 December 2021

Redirect to:

Invalid username
Login to AoPS