---
tag: 0DRF0F3XC9
kind: statement
role: theorem
name: Inequality of arithmetic and geometric means
aliases: [am-gm-inequality]
msc: ["26E60"]
sources:
  - url: https://en.wikipedia.org/wiki/AM%E2%80%93GM_inequality
    locator: The inequality
same_as:
  wikidata: Q841170
  wikipedia: AM–GM inequality
---

## Statement

Let $n \ge 1$ and let $x_1, \dots, x_n$ be nonnegative real numbers.
Then
$$\sqrt[n]{x_1 x_2 \cdots x_n} \le \frac{x_1 + x_2 + \dots + x_n}{n},$$
with equality if and only if $x_1 = x_2 = \dots = x_n$.

## Remarks

For $n = 2$, the inequality $\sqrt{x y} \le (x + y)/2$ follows from $(\sqrt{x} - \sqrt{y})^2 \ge 0$.
The weighted form: for weights $w_i \ge 0$ with $w_1 + \dots + w_n = 1$, $x_1^{w_1} \cdots x_n^{w_n} \le w_1 x_1 + \dots + w_n x_n$.

## Explanations

### L1 human

The geometric mean of some nonnegative numbers is never larger than their average.
For example, for $2$ and $8$, the geometric mean is $\sqrt{16} = 4$ and the average is $5$.
Among all rectangles with a fixed perimeter, the square has the largest area. This is the case $n = 2$.