Difference between revisions of "AM-GM inequality"
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| − | In [[algebra]], the '''AM-GM inequality''', | + | In [[algebra]], the '''AM-GM inequality''', sometimes called the '''inequality of arithmetic and geometric means''', 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 <math>x_1, x_2, \ldots, x_n \geq 0</math>, <cmath>\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n},</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. | In symbols, the inequality states that for any <math>x_1, x_2, \ldots, x_n \geq 0</math>, <cmath>\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n},</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. | ||
Revision as of 16:27, 27 November 2021
In algebra, the AM-GM inequality, sometimes called the inequality of arithmetic and geometric means, 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)
