Getting recursive formula to since solution

Question

Is there any way to get the recursive formula of the form $r_n=\alpha r_{n-1}+\beta$ to single formula as a function of $n$. I've seen results that find single formula as function of $n$ for geometric series and such but can't seem to find any literature or results for this type.

Answer 1

Hint: If $\alpha \ne 1$, ${r_n}=\alpha {r_{n-1}}+\beta \Rightarrow {r_n}+\frac{\beta }{\alpha -1}=\alpha \left( {r_{n-1}}+\frac{\beta }{\alpha -1} \right)$.