Harmonic Mean

Harmonic Mean

## Definition / Concept

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals.

## Formula

For positive numbers $x,y$:

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

If:

$$x,H,y$$

are in HP, then:

$$\frac1x,\frac1H,\frac1y$$

are in AP.

Therefore:

$$\frac2H=\frac1x+\frac1y$$

## Usage

- Used to insert harmonic means and solve problems involving APs of reciprocals.