I'm stuck on this exercise, I know I'm getting close, but I just can't see it

Prove by induction that $\displaystyle \sum_{i=0}^n i2^{i - 1} = (n -1)2^n + 1$.

Basically this boils down to showing that $\displaystyle ((k + 1) - 1)2^{k + 1} + 1 = (k - 1)2^k + 1 + (k + 1)2^{(k + 1) - 1$, simplified: $\displaystyle 2k^{k + 1} + 1 = (k - 1)2^k + 1 + (k + 1)2^k$

My right hand side simplifies to $\displaystyle (k - 1)2^k + 1 + (k + 1)2^k = 2k^k - 2^k + 1 + 2k^k + 2^k = 4k^k + 1 \ne 2k^{k + 1} + 1$

So obviously I'm missing something and I'm afraid it's as trivial as possible, but I just don't see it at the moment