However, it is possible for the characteristic polynomial to have only one root. This can happen if the characteristic polynomial factors as \((x - r)^2\text{.}\) It is still the case that \(r^n\) would be a solution to the recurrence relation, but we won't be able to find solutions for all initial conditions using the general form \(a_n = ar_1^n + br_2^n\text{,}\) since we can't distinguish between \(r_1^n\) and \(r_2^n\text{.}\) We are in luck though:
in-context