The inequality of arithmetic and geometric means

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Prove that for positive real numbers $x_{1}, \dots, x_{n}$

\frac{x _{1} + \dots + x _{n}}{n} \geq n x_{1} x_{2} \dots x_{n},

with equality only when all the $x_{i}$ are equal.

Posed by Augustin-Louis Cauchy, 1821

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algebra inequalities

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Cauchy's proof in his Cours d'analyse of 1821 goes forward to every power of two and then backward from $n$ to $n - 1$ . George Pólya later gave a proof in a few lines from the inequality $e^{t} \geq 1 + t$ . Dozens of other proofs are known.

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Inequality of arithmetic and geometric means

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