Recurrence relation with limit How do I find the lim for a series that has the following recurrence relation 
$$ a_{0}=2 $$ $$a_1=16 $$ and $$a_{n+1}^2=a_na_{n-1} $$ .I applied $\ln$ and I substituted $$ b_n=\ln (a_n) $$ and so I found the following recurrence relation $$2b_{n+1}=b_n+b_{n-1}$$ .What do I do next? Can someone explain this limit in recurrence relation method because I'm not very familiar with it. I understood that it can only be applied if the given series converges?

The Ansatz $b_n = x^n$ leads to $2x^2 - x - 1 = 0$ which implies $x = 1,-1/2$. Hence the general solution is $$b_n = c_1 + c_2 \left(-\frac{1}{2} \right)^n.$$
Using the initial conditions we see that $c_1 + c_2 = \ln 2$ and $c_1 - \frac{c_2}{2} = \ln 16$.
