Prove using mathematical induction that for every positive integer n, $\sum_{i=1}^n ( i * 2^i ) = (n-1) 2^{n+1} + 2 $

Question

Prove using mathematical induction that for every positive integer n,

$$\sum_{i=1}^ni2^i=(n-1) 2^{n+1} + 2$$

There is what i did so far :

Answer 1

Your solution is good, except the final form should be presented as the $k+1$ step, so it should look like

$$\sum_{i=1}^{k+1}i\cdot 2^i=\dots=k\cdot 2^{k+2}+2$$

(where the $\dots$ represent the necessary arithmetic in between.) This will make it clear that the final form is definitely the increment from the $k$th step to the $(k+1)$th step.