VMO 2022 Problem 1

Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as: $u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$ a) With $a=0$ , prove that there exist a finite limit of $(u_n)$ and find that limit b) With $a \ge 0$ , prove that there exist a finite limit of $(u_n)$