Problem about a recursive sequence $a_n = \sqrt{\frac{a_{n-1}}{a_{n-2}}}$ with initial conditions $a_0=8\text{ and }a_1=\frac{1}{2\sqrt{2}}$

Question

I am trying to solve the recurrence relation $$a_n = \sqrt{\frac{a_{n-1}}{a_{n-2}}}$$ with initial conditions $$a_0=8\text{ and }a_1=\frac{1}{2\sqrt{2}}$$

Can anyone please help me with the technique to approach this problem?

Answer 1

Take logs and it becomes $b_n = \dfrac{b_{n-1}+b_{n-2}}{2} $. This has been investigated here.

Note that $a_0a_1^2 = 1$. That is not a coincidence.