Question 1
Let $a$ be real and define
$$u_1=a,\qquad u_{n+1}=\frac12+\sqrt{\frac{2n+3}{n+1}u_n+\frac14}\quad(n\ge1).$$
a) For $a=5$, prove that the sequence has a finite limit and find it.
b) Find all $a$ for which the sequence is defined over the reals and has a finite limit.
$$u_1=a,\qquad u_{n+1}=\frac12+\sqrt{\frac{2n+3}{n+1}u_n+\frac14}\quad(n\ge1).$$
a) For $a=5$, prove that the sequence has a finite limit and find it.
b) Find all $a$ for which the sequence is defined over the reals and has a finite limit.