Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 7 / b / ii

The absorbing states give $e_0=e_n=0$. Multiplying the difference equation in (i) by $3$ givesThe second finite difference of $k^2$ is $2$, so a particular solution is $-3k^2/2$; the homogeneous solutions are $A+Bk$. Imposing the two boundary conditions gives $A=0,B=3n/2$. ThereforeUniqueness follows because the difference of any two solutions has zero second finite difference, hence is affine function, and vanishes at both endpoints. This is the mean absorption time of a lazy symmetric random walk with holding probability $1/3$.