# Harmonic Mean Formula explained

Learn and know what is the meaning of Harmonic Mean (H.M) and how to derive the Harmonic Mean Formula.

In sequence and series, we have three main topics i.e. Arithmetic Progression, Geometric Progression and Harmonic Progression. Arithmetic Mean (A.M), Geometric Mean (G.M) and Harmonic Mean (H.M) are the three formulas related to A.P, G.P and H.P which have some relation among them. We have direct formulas for Arithmetic Mean and geometric Mean. But there is no direct formula for Harmonic Mean. Now we will see what is the Harmonic Mean formula, and how to find it.

## Harmonic Mean Formula Derivation:

Let x, y, z be in Harmonic Progression. The middle term “y” is called as the Harmonic Mean. As we know that, If x, y, z be in Harmonic Progression then $\frac { 1 }{ x }$, $\frac { 1 }{ y }$, $\frac { 1 }{ z }$ are in A.P

By Arithmetic Mean definition,

$2\frac { 1 }{ y }$ = $\frac { 1 }{ x }$ + $\frac { 1 }{ z }$

$\frac { 2 }{ y }$ = $\frac { x+z }{ xy }$

$y=\frac { 2xy }{ x+z }$

Therefore, if x, y, z be in Harmonic Progression then the Harmonic Mean is given as, $y=\frac { 2xy }{ x+z }$

Hope you have understood the H.M derivation steps.