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

Result R4095 on D2454: Real arithmetic mean

Real arithmetic expression for real arithmetic mean

Formulation 0

Let $x_1, \dots, x_N \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \in \mathbb{R}$ is a D2454: Real arithmetic mean of $x_1, \dots, x_N$

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

Proofs

Proof 0

Let $x_1, \dots, x_N \in \mathbb{R}$ each be a D993: Real number such that

(i)	$a \in \mathbb{R}$ is a D2454: Real arithmetic mean of $x_1, \dots, x_N$

By hypothesis, $a$ satisfies \begin{equation} N a = \sum_{n = 1}^N a = \sum_{n = 1}^N x_n \end{equation} Diving each side by the positive integer $N \geq 1$, we obtain \begin{equation} a = \frac{1}{N} \sum_{n = 1}^N x_n \end{equation} $\square$