Characteristic equation of a difference equation indicates the function behavior

Question

For the characteristic equation $a_2 \lambda^2 + a_1 \lambda + a_0 = 0$ of the difference equation $a_2 x_{n+2} + a_1 x_{n+1} + a_0 x_n = 0$, I remember there is a way to indicate if the function of $f(n) = x_n$ is in/decreasing or oscillating? Thank you~

Answer 1

You will have $$f(n)=Ar^n+Bs^n$$ where $A$ and $B$ depend on $x_1$ and $x_2$, and where $r$ and $s$ are the roots of the characteristic equation. That should enable you to determine increasing, decreasing, and alternating.

Strictly speaking, I've left out the case of repeated roots. I can get back to that, if you want.