VWO 1986

Prove that for integer $n$ we have: $$n! \le \left( \frac{n+1}{2} \right)^n$$ (please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)

Let $\{a_k\}_{k\geq 0}$ be a sequence given by $a_0 = 0$ , $a_{k+1}=3\cdot a_k+1$ for $k\in \mathbb{N}$ .

Prove that $11 \mid a_{155}$