Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 7 / c / iii / Solution

The characteristic polynomial of the homogeneous linear recurrence is $(r-2{)}^2$, so its repeated-root solutions are $(A+Bn)2^n$. A constant particular solution is one. Thus $u_n=1+(A+Bn)2^n$. The initial conditions give $A=-1$ and $B=1$, henceThe factor $n$ is required by the repeated root of a polynomial; omitting it would lose one independent solution.