Question 90
Let $a_1 \gt \frac{1}{12}$ and $a_{n+1} = \sqrt{(n+2)a_n + 1}$ for $n \ge 1$. Prove that:
a) $a_n \gt n - \frac{2}{n}$;
b) the sequence $b_n = 2^n \left(\frac{a_n}{n} - 1\right)$, $n = 1, 2, \dots$, is convergent.
a) $a_n \gt n - \frac{2}{n}$;
b) the sequence $b_n = 2^n \left(\frac{a_n}{n} - 1\right)$, $n = 1, 2, \dots$, is convergent.