Recurrence and Inequality

Question

What is the standard (if it exists) way to solve a recurrence $$a_n\leq \alpha a_{n-1}+\beta a_{n-2}?$$

We can suppose $\alpha, \beta>0$ are positive and real.

Answer 1

For me the standard way is applied when we are given non-negative $a_0$ and $a_1$. Then for each $n$ we have $a_n\le b_n$, where $b_0=a_0$, $b_1=a_1$, and $b_n=\alpha b_{n-1}+\beta b_{n-2}$ for each $n\ge 2$, and this bound is tight. The recurrence for $b_n$ is order two homogeneous linear recurrence with constant coefficients, which can be solved via the standard way.