Rewrite the recurrence relation \(a_n - 7a_{n-1} + 10a_{n-2} = 0\text{.}\) Now form the characteristic equation:

\begin{equation*} x^2 - 7x + 10 = 0 \end{equation*}

and solve for \(x\text{:}\)

\begin{equation*} (x - 2) (x - 5) = 0 \end{equation*}

so \(x = 2\) and \(x = 5\) are the characteristic roots. We therefore know that the solution to the recurrence relation will have the form

\begin{equation*} a_n = a 2^n + b 5^n\text{.} \end{equation*}