VMO 2017 Problem 1

Given $a\in\mathbb{R}$ and a sequence $(u_n)$ defined by $$\begin{cases} u_1=a u_{n+1}=\frac{1}{2}+\sqrt{\frac{2n+3}{n+1}u_n+\frac{1}{4}} \forall n\in\mathbb{N}^* \end{cases}$$

a) Prove that $(u_n)$ is convergent sequence when $a=5$ and find the limit of the sequence in that case

b) Find all $a$ such that the sequence $(u_n)$ is exist and is convergent.