Inequality of arithmetic and geometric means
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In algebra, the inequality of arithmetic and geometric means, or the AM–GM inequality, states that the arithmetic mean is greater than or equal to the geometric mean of any list of nonnegative reals; furthermore, equality holds if and only if every real in the list is the same.
In symbols, the inequality states that for any 
, ![\[\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n},\]](http://latex.artofproblemsolving.com/d/8/5/d853590d8071c60b02aa630b5530eb94b2041cda.png)
with equality if and only if 
.
NOTE: This article is a work-in-progress and meant to replace the Arithmetic mean-geometric mean inequality article, which is of poor quality.
OUTLINE:
- Proofs
- Links to Proofs of AM-GM Inequality
- Generalizations
- Weighted AM-GM
- QM-AM-GM-HM (with or without weights)
- Power Mean (with or without weights)
