Rewrite the recurrence relation \(a_n - 7a_{n-1} + 10a_{n-2} = 0\text{.}\) Now form the characteristic equation:
and solve for \(x\text{:}\)
so \(x = 2\) and \(x = 5\) are the characteristic roots. We therefore know that the solution to the recurrence relation will have the form
To find \(a\) and \(b\text{,}\) plug in \(n =0\) and \(n = 1\) to get a system of two equations with two unknowns:
Solving this system gives \(a = \frac{7}{3}\) and \(b = -\frac{1}{3}\) so the solution to the recurrence relation is