Solution
= 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 $B=1$, hence
$$
\boxed{u_n=1+(n-1)2^n.}
$$
The factor $n$ is required by the repeated <root of a polynomial>; omitting it would lose one independent solution.