harmonic progression

Elements
a₁ = value of the first term
a_m = value of any term after the first term but before the last term
a_n = value of the last term
n = total number of terms
m = m^th term after the first but before n^th
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum of the 1^st n terms
 

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.