Question 142
Let $a_{0}, a_{1}, a_{2}, \ldots$ be an infinite strictly increasing sequence of positive integers such that for each $n \geqslant 1$ we have
$$
a_{n} \in\left\{\frac{a_{n-1}+a_{n+1}}{2}, \sqrt{a_{n-1} \cdot a_{n+1}}\right\}
$$
Let $b_{1}, b_{2}, \ldots$ be an infinite sequence of letters defined as
$$
b_{n}= \begin{cases}A, & \text{ if } a_{n}=\frac{1}{2}\left(a_{n-1}+a_{n+1}\right) \\ G, & \text{ otherwise }\end{cases}
$$
Prove that there exist positive integers $n_{0}$ and $d$ such that for all $n \geqslant n_{0}$ we have $b_{n+d}=b_{n}$.
$$
a_{n} \in\left\{\frac{a_{n-1}+a_{n+1}}{2}, \sqrt{a_{n-1} \cdot a_{n+1}}\right\}
$$
Let $b_{1}, b_{2}, \ldots$ be an infinite sequence of letters defined as
$$
b_{n}= \begin{cases}A, & \text{ if } a_{n}=\frac{1}{2}\left(a_{n-1}+a_{n+1}\right) \\ G, & \text{ otherwise }\end{cases}
$$
Prove that there exist positive integers $n_{0}$ and $d$ such that for all $n \geqslant n_{0}$ we have $b_{n+d}=b_{n}$.