算术——几何平均不等式是一个有着广泛应用的重要不等式．证明这个不等式有多种方法，但都较繁．本文给出一个比较简捷的证明方法．定理设ａ１，ａ２，…，ａｎ是ｎ个（ｎ∈Ｎ且ｎ≥２）正数，则１ｎ（ａ１＋ａ２＋…＋ａｎ）≥ｎａ１ａ２…ａｎ，当且仅当ａ１＝ａ２＝…... Arithmetic - geometric mean inequality is an important inequality with wide application. There are many ways to prove this inequality, but they are more complicated. This article gives a simpler proof method. Theorem Let a1, a2,...,an be n (n∈N and n≥2) positive numbers, then 1n(a1+a2+...+an)≥na1a2...an if and only if a1=a2=...