Nonlinear recurrence relation

Question

I have a question about solving nonlinear recurrence relation.

$(a_{n+1}*a_n)^2 = \sqrt{a_n} * 2^n$ and $a_0=1$

I do not know how to solve this kind of relations. What is the strategy to solve them? Thanks for your help.

Answer 1

Hint: If your relation is $a_{n+1}^2a_n^2=2^n \sqrt{a_n}$, then put $a_n=2^{u_n}$, with $u_1=0$. If I am not wrong, you will find that $u_{n+1}+\frac{3}{4}u_n=\frac{n}{2}$, that is easy to solve. (Equivalently, you can look at $b_n=\log a_n$).