Category:Cauchy's Mean Theorem

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This category contains pages concerning Cauchy's Mean Theorem:


Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.

Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.

Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.


Then:

$A_n \ge G_n$

with equality holding if and only if:

$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$

That is, if and only if all terms are equal.