Question 1
Define a sequence $(a_n)$ as follows
$$
\begin{cases} a_1 = 1, \\ a_{n+1} = 3 - \frac{a_n + 2}{2^{a_n}} \quad \text{for } n \ge 1. \end{cases}
$$
Prove that this sequence has a finite limit and find this limit.
$$
\begin{cases} a_1 = 1, \\ a_{n+1} = 3 - \frac{a_n + 2}{2^{a_n}} \quad \text{for } n \ge 1. \end{cases}
$$
Prove that this sequence has a finite limit and find this limit.