ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4118 on D2455: Real geometric mean

Real arithmetic expression for unsigned real geometric mean

Formulation 0

Let $x_1, \ldots, x_N \in [0, \infty)$ each be a D993: Real number such that

(i)	$a \in \mathbb{R}$ is a D2455: Real geometric mean of $x_1, \ldots, x_N$

Then \begin{equation} a = \left( \prod_{n = 1}^N x_n \right)^{1 / N} \end{equation}

Formulation 1

Let $x_1, \ldots, x_N \in [0, \infty)$ each be a D993: Real number such that

(i)	$a \in \mathbb{R}$ is a D2455: Real geometric mean of $x_1, \ldots, x_N$

Then \begin{equation} a = (x_1 x_2 \cdots x_N)^{1 / N} \end{equation}

Proofs

Proof 0

Let $x_1, \ldots, x_N \in [0, \infty)$ each be a D993: Real number such that

(i)	$a \in \mathbb{R}$ is a D2455: Real geometric mean of $x_1, \ldots, x_N$

By hypothesis, $a$ satisfies \begin{equation} a^N = \prod_{n = 1}^N a = \prod_{n = 1}^N x_n \end{equation} Raising both sides to power $1/N$, we obtain \begin{equation} a = (a^N)^{1/N} = \left( \prod_{n = 1}^N x_n \right)^{1/N} \end{equation} $\square$