Question 6
Let $ a_0, a_1, a_2, \ldots $ be an infinite strictly increasing sequence of positive integers such that for each $ n \geq 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
$$ b_{n}=\begin{cases}A,&\text{if} a_{n}=\frac{1}{2}(a_{n-1}+a_{n+1});\\G,&\text{otherwise}.\end{cases} $$
Prove that there exist positive integers $ n_{0} $ and d such that for all $ n \geq 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
$$ b_{n}=\begin{cases}A,&\text{if} a_{n}=\frac{1}{2}(a_{n-1}+a_{n+1});\\G,&\text{otherwise}.\end{cases} $$
Prove that there exist positive integers $ n_{0} $ and d such that for all $ n \geq n_{0} $ we have $ b_{n+d} = b_{n} $.