An Induction Problem, What Am I Supposed To Prove?

You are supposed to prove $u_n=3(2^n)+(-1)^n$.

$u_1=5$ and $u_{n+1}=2u_n-3(-1)^n$ are the conditions you are supposed to make use of.

$u_{n+1}=2u_n-3(-1)^n$ is your recurrence, specifying on how to obtain the $n+1$-st term $u_{n+1}$ from the $n$-th term $u_n$. Together with a start value, $u_1=5$, this stepwise determines the sequence completely.

What the problem is trying to establish, is to show that in general, for any $n$, you can obtain the $n$-th term directly(without evaluating all $u_k$'s with $k<n$ before) via the formula $u_n=3(2^n)+(-1)^n$. That this formula holds can be shown using induction.