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If \(r_1\) and \(r_2\) are two distinct roots of the characteristic polynomial (i.e., solutions to the characteristic equation), then the solution to the recurrence relation is

\begin{equation*} a_n = ar_1^n + br_2^n\text{,} \end{equation*}

where \(a\) and \(b\) are constants determined by the initial conditions.

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