Year 12s: what will be the first way you research your future options? >>

Demonstrating/deducting the Harmonic and Geomtric Means

Hello everyone,

At friday my Statistics' professor challenged us to deduce both Harmonic and Geometric Means formula.
I looked over the internet but couldn't find anything related to this, and I don't have any idea how to start this demonstration. I never saw any demonstration of any Means: in all my life they just showed me the formula and taught how to use and interpretate it.
I used to think that it was some kind of formula they just came up with, without having any theory behind, just a definition. But now that I stopped to think about it a little I saw how absurd this sounds like: there must be an argument behind this Means, and so there must be a way to demonstrate both formulas using or Algebra or Geometrics, or anything like that!
Do you guys know how to do it?

(edited 10 years ago)

Reply 1

Smaug123

Original post by negativer

…and so there must be a way to demonstrate both formulas using or Algebra or Geometrics, or anything like that!
Do you guys know how to do it?

There's an interpretation to do with triangles (https://en.wikipedia.org/wiki/Pythagorean_means).
For the arithmetic mean, I would say something along the lines of "The point that is 'in the middle of' two points on a line is (a+b)/2; this generalises in the obvious way". The geometric mean comes about from considering the exponential of the arithmetic mean.

Reply 2

negativer

Original post by Smaug123

The arithmetic I understood, it really makes sense thinking in this way. But the geometric mean I still can't understood: what significance does the exponential of the Arithmetic mean has? I mean, it is defined as the exponential of the Arithmetic mean, purely like that? Or it has a geometric foundation as the Arithmetic does?
And what about the Harmonic?
In the link you sent me I saw a figure that ilustrates a geometric construction of some Mathematical means: https://en.wikipedia.org/wiki/File:MathematicalMeans.svg
But how can I, starting on this figure, reach the formulas of Geometric and Harmonic means?

Reply 3

negativer

Right! I understand now the arithmetic mean.
But I'm still having troubles with the Geometric. I can't just say that it is the exponentian of the arithmetic, can I? I mean, what's the foundation on this demonstration? I was thinking in a demonstration/definition just as the Arithmetics one you gave. And plus, what about the harmonic?

In the link you sent me I saw this image:
800px-MathematicalMeans.svg.png

Can this be used to demonstrate the Harmonic and Geometric Means?
What I'm having doubts is in the generalisation of the formula: this image is for two values, and i can't think a way to generalise this.

Reply 4

Smaug123

Well, one of the reasons there are so many different means is that there are various ways to generalise the formula. They actually all follow from the power mean:
https://en.wikipedia.org/wiki/Generalized_mean
To be honest, I don't know how you'd come up with that in advance.