Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 1A / iii / Solution

The forcing base $-2$ is a root of the characteristic equation of a linear recurrence, so a constant multiple of $(-2{)}^n$ cannot be a particular solution. For $P(r)=r^3-3r+2$, direct substitution givesFor the resonant case of exponential forcing in a linear recurrence, at $r=-2$, $P(-2)=0$ and $(-2)P^{\prime }(-2)=-18$. Hence $-(n/18)(-2{)}^n$ is a particular solution of the inhomogeneous linear recurrence. The general solution is