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Definition D3306
Real harmonic mean

Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in (0, \infty)$ is a harmonic mean of $x_1, \dots, x_N$ if and only if $$\sum_{n = 1}^N \frac{1}{a} = \sum_{n = 1}^N \frac{1}{x_n}$$

Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in (0, \infty)$ is a harmonic mean of $x_1, \dots, x_N$ if and only if $$\underbrace{\frac{1}{a} + \frac{1}{a} + \cdots + \frac{1}{a}}_{N \text{ times}} = \frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_N}$$

Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in (0, \infty)$ is a harmonic mean of $x_1, \dots, x_N$ if and only if $$\sum_{n = 1}^N a^{-1} = \sum_{n = 1}^N x^{-1}_n$$
Results
 ▶ R4092: Real arithmetic expression for real harmonic mean ▶ R4093: Real harmonic and arithmetic means are multiplicative inverses when arguments are