Question 168
Given a real number $a$ and consider the sequence $(u_n)$ defined by
$$
u_1 = a, \quad u_{n+1} = \frac{1}{2} + \sqrt{\frac{2n+3}{n+1}u_n + \frac{1}{4}}, \quad \forall n \in \mathbb{N}^*.$$
1.
If $a = 5$, prove that $(u_n)$ has a finite limit and find that limit.
2.
Find all the values of $a$ such that the sequence $(u_n)$ is well defined and has a finite limit.
$$
u_1 = a, \quad u_{n+1} = \frac{1}{2} + \sqrt{\frac{2n+3}{n+1}u_n + \frac{1}{4}}, \quad \forall n \in \mathbb{N}^*.$$
1.
If $a = 5$, prove that $(u_n)$ has a finite limit and find that limit.
2.
Find all the values of $a$ such that the sequence $(u_n)$ is well defined and has a finite limit.