# Intuition and derivation of the geometric mean

I've run through a bunch of searches, especially here on SO, but I simply couldn't find something that answers a question that has been on my mind lately. How was the geometric mean derived? What is the intuition behind it. Most simply use the final equation as a justification for its existence.

The $n$-th root of the product of all elements equation really doesn't do it justice.

Could someone elaborate on how and why is it interesting?

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The phrase "why it works" doesn't make sense. Do you mean "why is it interesting"? –  KCd Apr 29 '12 at 21:00
@KCd Yes, that is acceptable. I just wish some intuition about it, how it was derived etc. –  Rebuq Apr 29 '12 at 21:03

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