Mean Inequality for N Terms

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Demonstrate the inequality of averages for a quantity of $N$ terms.

I was thinking of a geometric demonstration of this, like $ n = 2$

This is just an image I found on the internet that illustrates inequality from geometry, but to follow my resolution:

A triangle inscribed in a circle with radii $ a $ and $ b $ and height $ h $

$2R = a+b$

$R= \frac{a+b}{2}$ (AM)

and

$h^2=ab$

$h+\sqrt{ab}$

enter image description here