Cauchy-Schwarz: Problema 19

Let \(a_{1}, \ldots, a_{n} >0\) and \(n > 12\) be such that \(
\sum_{k=1}^{n} a_{k}=1\) and \( \sum_{k=1}^{n} ka_{k} = 2.\) Prove
that: $$ (a_{2}-a_{1})\sqrt{2} + (a_{3}-a_{2})\sqrt{3}+ \ldots +
(a_{n}-a_{n-1})\sqrt{n} < 0.$$

Gabriel Dospinescu