Question 1
Let $a\ge0$ be real and define
$$u_1=6,\qquad u_{n+1}=\frac{2n+a}{n}+\sqrt{\frac{n+a}{n}u_n+4}\quad(n\ge1).$$
(a) For $a=0$, prove that $(u_n)$ has a finite limit and determine it.
(b) For every $a\ge0$, prove that $(u_n)$ has a finite limit.
$$u_1=6,\qquad u_{n+1}=\frac{2n+a}{n}+\sqrt{\frac{n+a}{n}u_n+4}\quad(n\ge1).$$
(a) For $a=0$, prove that $(u_n)$ has a finite limit and determine it.
(b) For every $a\ge0$, prove that $(u_n)$ has a finite limit.