The arithmetic mean–geometric mean (AM–GM) inequality states that for positive real numbers \(x_1\), \(x_2\), …, \(x_n\), \[\frac{1}{n}\sum_{i=1}^{n} x_{i} \geq \Bigl(\prod_{i=1}^{n} x_{i}\Bigr)^{\frac{1}{n}},\] that is, the arithmetic mean of a set is always greater than or equal to its geometric mean, with equality if and only if all of the \(x_i\) are equal.
For example, the case \(n=2\) states that if we have two positive numbers \(x\) and \(y\), they satisfy: \[\frac{x+y}{2} \geq \sqrt{\vphantom{()}x y}\] with equality if and only if \(x=y\).
