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Darsh Ranjan
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Here's a proof of the inequality of the arithmetic and geometric means in the form $$\frac{x\_1^n}{n} + \cdots + \frac{x\_n^n}{n} \geq x\_1\cdots x\_n.$$

Proof for $n=3$:

(there should be a figure here...) http://img64.imageshack.us/img64/5738/arithgeom01b.png

The "figure" for general $n$ is similar, with $n$ right pyramids, one with an $(n-1)$-cube of side length $x\_k$ as its base and height $x\_k$ for each $k=1,\ldots,n$.

(I made this in Inkscape, a wonderful free-software vector drawing application. For the inequality and associated labels, I used the textext extension.)

Darsh Ranjan
  • 6k
  • 2
  • 52
  • 57