Exercise 8.

Suppose that \(r^n\) and \(q^n\) are both solutions to a recurrence relation of the form \(a_n = \alpha a_{n-1} + \beta a_{n-2}\text{.}\) Prove that \(c\cdot r^n + d \cdot q^n\) is also a solution to the recurrence relation, for any constants \(c, d\text{.}\)