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 \begin{equation} \sum_{n = 1}^N \frac{1}{a} = \sum_{n = 1}^N \frac{1}{x_n} \end{equation}

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 \begin{equation} \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} \end{equation}

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 \begin{equation} \sum_{n = 1}^N a^{-1} = \sum_{n = 1}^N x^{-1}_n \end{equation}