Proof by induction, induction step

I am trying to prove

$$\sum_{k=1}^n k2^{k-1} = 1+(n-1)2^n$$

I proved the base case with $n = 1$. I am having trouble proving the induction step.

I know I need to prove for $n = n +1$ so I got

$$\sum_{k=1}^n k2^{k-1}+(k+1)2^{(k+1)-1} = 1+[(n+1)-1]2^{n+1}$$

suppose $n = i$

$$1 + [(i+1)-1] 2^{i+1}$$

I am not sure if I am on the right path and how to simplify from here onwards? Could someone point me in the right direction?