Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

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For two numbers x and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If x, a, y is a geometric progression then 'a' is called geometric mean. If x, a, y form a harmonic progression then 'a' is called harmonic mean.
 

Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula
 

$ AM \times HM = GM^2 $

 

Below is the derivation of this relationship.
 

Derivation of AM × HM = GM2
$ x, \, AM, \, y $   →   arithmetic progression

Taking the common difference of arithmetic progression,
$ AM - x = y - AM $

$ y = 2 \, AM - x $   →   Equation (1)
 

$ x, \, GM, \, y $   →   geometric progression

The common ratio of this geometric progression is
$ \dfrac{GM}{x} = \dfrac{y}{GM} $

$ y = \dfrac{GM^2}{x} $   →   Equation (2)
 

Equate Equations (1) and (2)
$ 2 \, AM - x = \dfrac{GM^2}{x} $

$ 2x \, AM - x^2 = GM^2 $

$ x^2 - 2x \, AM = -GM^2 $   →   Equation (3)
 

$ x, \, HM, \, y $   →   harmonic progression

$ \dfrac{1}{x}, \, \dfrac{1}{HM}, \, \dfrac{1}{y} $   →   the reciprocal of each term will form an arithmetic progression

The common difference is
$ \dfrac{1}{HM} - \dfrac{1}{x} = \dfrac{1}{y} - \dfrac{1}{HM} $

$ \dfrac{2}{HM} - \dfrac{1}{x} = \dfrac{1}{y} $

$ \dfrac{2x - HM}{x \, HM} = \dfrac{1}{y} $

$ y = \dfrac{x \, HM}{2x - HM} $   →   Equation (4)
 

Equate Equations (1) and (4)
$ 2 \, AM - x = \dfrac{x \, HM}{2x - HM} $
$ (2 \, AM - x)(2x - HM) = x \, HM $
$ 4x \, AM - 2 \, AM \, HM - 2x^2 + x \, HM = x \, HM $
$ 4x \, AM - 2 \, AM \, HM - 2x^2 = 0 $
$ 2x \, AM - AM \, HM - x^2 = 0 $
$ x^2 - 2x \, AM = -AM \, HM $   →   Equation (5)
 

Equate Equations (3) and (5)
$ -AM \, HM = -GM^2 $

$ AM \times HM = GM^2 $

 



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Re: Relationship Between Arithmetic Mean, Harmonic Mean, and ...

Submitted by Andrew (guest) on May 4, 2011 - 4:41am.

Please see the following URL for a plain English description of these three averages as well as helpful tips on when to use each.

http://economistatlarge.com/finance/applied-finance/differences-arithmetic-geometric-harmonic-means

Re: Relationship Between Arithmetic Mean, Harmonic Mean, and ...

Submitted by Romel on May 13, 2011 - 10:00am.

Hello Andrew, you've got a cool very informative website. Good luck.