Question 68
Let $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$ be infinite sequences of real numbers satisfying $a_{n+1} + b_{n+1} = \frac{a_n + b_n}{2}$ and $a_{n+1}b_{n+1} = \sqrt{a_n b_n}$ for all $n \ge 1$. Suppose $b_{2016} = 1$ and $a_1 \gt 0$. Find all possible value(s) of $a_1$.