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
Arithmetic Progression

$ x, \, AM, \, y $   →   arithmetic progression
 

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

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

 

Geometric Progression

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

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

$ xy = GM^2 $   →   Equation (2)

 

Harmonic Progression

$ 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}{y} + \dfrac{1}{x} $

$ \dfrac{2}{HM} = \dfrac{x + y}{xy} $   →   Equation (3)

 

Substitute x + y = 2AM from Equation (1) and xy = GM2 from Equation (2) to Equation (3)

$ \dfrac{2}{HM} = \dfrac{2 \, AM}{GM^2} $

$ GM^2 = AM \times HM $   →   Okay!

 

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