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Arithmetic, Geometric, and Harmonic Means

An arithmetic mean is the value \(a\) such that:

$$ x_1+x_2+\cdots+x_n = \underbrace{a+a+\cdots+a}_{n} $$

A geometric mean is the value \(g\) such that:

$$ x_1 \times x_2 \times \cdots \times x_n = \underbrace{g \times g \times \cdots \times g}_{n} $$

A harmonic mean is the value \(h\) such that:

$$ \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}= \underbrace{\frac{1}{h}+\frac{1}{h}+\cdots+\frac{1}{h}}_{n} $$

Deriving the standard formulas

Arithmetic mean:

$$ \begin{aligned} \underbrace{a+a+\cdots+a}_{n} &= x_1+x_2+\cdots+x_n \\ na &= x_1+x_2+\cdots+x_n \\ a &= \frac{x_1+x_2+\cdots+x_n}{n} \end{aligned} $$

Geometric mean:

$$ \begin{aligned} \underbrace{g \times g \times \cdots \times g}_{n} &= x_1 \times x_2 \times \cdots \times x_n \\ g^n &= x_1 \times x_2 \times \cdots \times x_n \\ g &= \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n} \end{aligned} $$

Harmonic mean: $$ \begin{aligned} \underbrace{\frac{1}{h}+\frac{1}{h}+\cdots+\frac{1}{h}}_{n} &= \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n} \\ \frac{n}{h} &= \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n} \\ h &= \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}} \\ h &= \frac{n}{ \displaystyle \sum_{i=1}^n \frac{1}{x_i}} \end{aligned} $$