Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers
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
Below is the derivation of this relationship.
Derivation of AM × HM = GM2
Arithmetic Progression
Taking the common difference of arithmetic progression,
$AM - x = y - AM$
$x + y = 2 \, AM$ ← Equation (1)
Geometric Progression
The common ratio of this geometric progression is
$\dfrac{GM}{x} = \dfrac{y}{GM}$
$xy = GM^2$ ← Equation (2)
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)
$GM^2 = AM \times HM$ ← Okay!
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