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
Taking the common difference of arithmetic progression,
→ Equation (1)
The common ratio of this geometric progression is
→ Equation (2)
→ the reciprocal of each term will form an arithmetic progression
The common difference is
→ Equation (3)
Substitute x + y = 2AM from Equation (1) and xy = GM2 from Equation (2) to Equation (3)