Question 190
Let $a$ be a non-negative real number and let the sequence $(u_n)$ be defined by
$$
u_1 = 6, \quad u_{n+1} = \frac{2n + a}{n} + \sqrt{\frac{n+a}{n}u_n + 4}
$$
for all positive integers $n$.
a) For $a = 0$, prove that $(u_n)$ has a finite limit and find its value.
b) For $a \ge 0$, prove that $(u_n)$ has a finite limit.
$$
u_1 = 6, \quad u_{n+1} = \frac{2n + a}{n} + \sqrt{\frac{n+a}{n}u_n + 4}
$$
for all positive integers $n$.
a) For $a = 0$, prove that $(u_n)$ has a finite limit and find its value.
b) For $a \ge 0$, prove that $(u_n)$ has a finite limit.