Question 88
A sequence $\{a_{n}\}_{n \geq 1}$ of positive reals is defined by the rule $a_{n+1} a_{n-1}^{5} = a_{n}^{4} a_{n-2}^{2}$ for integers $n \gt 2$ together with the initial values $a_{1} = 8$, $a_{2} = 64$, and $a_{3} = 1024$. Compute
$$
\sqrt{a_{1} + \sqrt{a_{2} + \sqrt{a_{3} + \cdots}}}
$$
$$
\sqrt{a_{1} + \sqrt{a_{2} + \sqrt{a_{3} + \cdots}}}
$$