(a) Consider the homogeneous $k$ th-order difference equation

$$a_{k} y_{n+k}+a_{k-1} y_{n+k-1}+\ldots+a_{2} y_{n+2}+a_{1} y_{n+1}+a_{0} y_{n}=0$$

where the coefficients $a_{k}, \ldots, a_{0}$ are constants. Show that for $\lambda \neq 0$ the sequence $y_{n}=\lambda^{n}$ is a solution if and only if $p(\lambda)=0$, where

$$p(\lambda)=a_{k} \lambda^{k}+a_{k-1} \lambda^{k-1}+\ldots+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}$$

State the general solution of $(*)$ if $k=3$ and $p(\lambda)=(\lambda-\mu)^{3}$ for some constant $\mu$.

(b) Find an inhomogeneous difference equation that has the general solution

$$y_{n}=a 2^{n}-n, \quad a \in \mathbb{R}$$