Harmonic Progression

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression.

If the sequence:

$$ a_1, a_2, a_3, \dots $$

is a harmonic progression, then:

$$ \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots $$

forms an arithmetic progression.

If the reciprocal sequence follows an AP:

$$ \frac{1}{a_n} = \frac{1}{a} + (n - 1)d $$

Then the nth term of the HP is:

$$ a_n = \frac{1}{\frac{1}{a} + (n - 1)d} $$

The harmonic mean of two numbers $a$ and $b$ is given by:

$$ \text{HM} = \frac{2ab}{a + b} $$

The harmonic mean is useful in problems involving average speeds, resistances in parallel circuits, and rates.

The terms of an HP are always positive if all terms of the AP formed by reciprocals are defined.
Unlike AP or GP, you rarely sum the terms of an HP — the focus is usually on identifying or inserting harmonic means.
To insert harmonic means between two numbers, insert arithmetic means between their reciprocals, then take the reciprocal of each result.

Given any two positive numbers $a$ and $b$, we can define the following:

$$ \text{HM} \le \text{GM} \le \text{AM} $$

Equality holds only when $a = b$.

AM is useful for equal sharing (e.g., average grade).
GM is used when growth or proportion is involved (e.g., interest rates, scaling).
HM applies when the quantity being averaged is a rate (e.g., average speed, resistance).