Characteristic Root Technique for Repeated Roots.
Suppose the recurrence relation \(a_n = \alpha a_{n-1} + \beta a_{n-2}\) has a characteristic polynomial with only one root \(r\text{.}\) Then the solution to the recurrence relation is
\begin{equation*}
a_n = ar^n + bnr^n
\end{equation*}
where \(a\) and \(b\) are constants determined by the initial conditions.