Mathematics and Statistics · Ch 4 — Sequences and Series

Arithmetic, Geometric and Harmonic Means

For two positive numbers $a$ and $b$ , three different "averages" arise naturally — one from each progression.

Note

The Three Means of $a$ and $b$

Arithmetic Mean (AM): the number $A$ such that $a, A, b$ is an AP $\Rightarrow$ $A = \frac{a + b}{2}.$
Geometric Mean (GM): the number $G$ such that $a, G, b$ is a GP $\Rightarrow$ $G = \sqrt{ab}.$
Harmonic Mean (HM): the number $H$ such that $a, H, b$ is an HP $\Rightarrow$ $H = \frac{2ab}{a + b}.$

The HM formula follows from the HP rule: $\tfrac1a, \tfrac1H, \tfrac1b$ must be an AP, so $\tfrac1H = \tfrac12\left(\tfrac1a + \tfrac1b\right) = \tfrac{a+b}{2ab}$ , giving $H = \tfrac{2ab}{a+b}$ .

Note

Two Key Relationships

$G^2 = A \times H$ — the geometric mean is itself the geometric mean of the arithmetic and harmonic means.
$A \ge G \ge H$ for all positive $a, b$ , with equality only when $a = b$ . …

Definition 11Arithmetic Mean (AM)

For two numbers $a, b$ : $A = \dfrac{a+b}{2}$ ; the middle term making $a, …

Definition 12Geometric Mean (GM)

For two positive numbers $a, b$ : $G = \sqrt{ab}$ ; the middle term making $ …

Definition 13Harmonic Mean (HM)

For two positive numbers $a, b$ : $H = \dfrac{2ab}{a+b}$ ; the middle term making $a, H, b$ an HP. Satisfies $G^2 = A\cdot H …

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