For two numbers [math]x[/math] and [math]y[/math], let [math]\,x, \, a, \, y\,[/math] be a sequence of three numbers. If [math]\,x, \, a, \, y\,[/math] is an arithmetic progression then [math]a[/math] is called arithmetic mean. If [math]\,x, \, a, \, y\,[/math] is a geometric progression then [math]a[/math] is called geometric mean. If [math]\,x, \, a, \, y\,[/math] form a harmonic progression then [math]a[/math] is called harmonic mean.
Let [math]AM[/math] = arithmetic mean, [math]GM[/math] = geometric mean, and [math]HM[/math] = harmonic mean. The relationship between the three is given by the formula
Below is the derivation of this relationship.
[math]x, \, AM, \, y \, \to \,[/math] arithmetic progression
Taking the common difference of arithmetic progression,
[math]AM - x = y - AM[/math]
[math]y = 2 \, AM - x \, \to \, [/math] Equation (1)
[math]x, \, GM, \, y \, \to \,[/math] geometric progression
The common ratio of this geometric progression is
[math]\dfrac{GM}{x} = \dfrac{y}{GM}[/math]
[math]y = \dfrac{GM^2}{x} \, \to \, [/math] Equation (2)
Equate Equations (1) and (2)
[math]2 \, AM - x = \dfrac{GM^2}{x}[/math]
[math]2x \, AM - x^2 = GM^2[/math]
[math]x^2 - 2x \, AM = -GM^2 \, \to \, [/math] Equation (3)
[math]x, \, HM, \, y \, \to \,[/math] harmonic progression
[math]\dfrac{1}{x}, \, \dfrac{1}{HM}, \, \dfrac{1}{y} \, \to \,[/math] the reciprocal of each term will form an arithmetic progression
The common difference is
[math]\dfrac{1}{HM} - \dfrac{1}{x} = \dfrac{1}{y} - \dfrac{1}{HM}[/math]
[math]\dfrac{2}{HM} - \dfrac{1}{x} = \dfrac{1}{y}[/math]
[math]\dfrac{2x - HM}{x \, HM} = \dfrac{1}{y}[/math]
[math]y = \dfrac{x \, HM}{2x - HM} \, \to \, [/math] Equation (4)
Equate Equations (1) and (4)
[math]2 \, AM - x = \dfrac{x \, HM}{2x - HM}[/math]
[math](2 \, AM - x)(2x - HM) = x \, HM[/math]
[math]4x \, AM - 2 \, AM \, HM - 2x^2 + x \, HM = x \, HM[/math]
[math]4x \, AM - 2 \, AM \, HM - 2x^2 = 0[/math]
[math]2x \, AM - AM \, HM - x^2 = 0[/math]
[math]x^2 - 2x \, AM = -AM \, HM \, \to \, [/math] Equation (5)
Equate Equations (3) and (5)
[math]-AM \, HM = -GM^2[/math]