Sum of squares compared to square of sum, divided by number of elements being summed

Question

Let $M, S > 0$ such that $M < S$, where $M$ is a positive integer and $S = \sum_{k = 1}^{M} a_k$, and each $a_k$ is a positive integer. Is it always the case that $\frac{S^2}{M} = \frac{\big(\sum_{k = 1}^{M} a_k\big)^2}{M} \leq \sum_{k = 1}^{M} a_k^2$?

Answer 1

Your claimed identity may be rearranged as $\frac1M\sum_{k=1}^Ma_k\le\sqrt{\frac{\sum_ka_k^2}{M}}$. This is called the RMS-AM inequality or QM-AM inequality. It's equivalent to the fact that the variance of the $a_k$ is non-negative.