Question 130
Prove the following inequality for all $a_{1}, \ldots, a_{n}\gt 0$ and determine all cases in which the equal sign appears:
$$
\sum_{k=1}^{n} k a_{k} \leq \binom{n}{2} + \sum_{k=1}^{n} a_{k}^{k}
$$
$$
\sum_{k=1}^{n} k a_{k} \leq \binom{n}{2} + \sum_{k=1}^{n} a_{k}^{k}
$$