Is it possible to determine whether the solutions of a second order difference equation with variable coefficients $$x_{n+2}+a_n x_{n+1}+b_n x_{n}=0$$ are growing with $n$ (for example exponentially) or not ? If yes, is it possible to determine the growth rate?
Edit: In response to the comment of "Did" you are right it depends on $a_n$ and $b_n$, what I had in mind was this: Is there any criteria for deciding whether the solutions are growing or not depending on properties of these coefficients without actually solving the equation?