MK-MO 2021 Problem 1

Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$ , prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$ .