A Weighted Sum's Extremes

Problem

Let $n \geqslant 2$ and $a_1, a_2, \ldots, a_n \geqslant 0$ with $\displaystyle\sum_{i=1}^n a_i^2 = S$ . Find, in terms of $S$ and $n$ , the maximum and minimum values of

M = \sum_{i=1}^n i\,a_i.

Answer

Solution

Difficulty	7/10
Topics	algebra, Cauchy-Schwarz, inequality

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