prove by mathematical induction

Question

I've been trying to solve this but I'm having trouble in simplifying it, in order to match the right hand side. Could you solve this? $$\sum_{i=1}^{n+1} i\cdot 2^i = n\cdot 2^{n+2} +2 ,$$ for all integers $n\ge 0$.

Answer 1

$n2^{n+2}+2= \sum_{i\leq n+1} i2^î= \sum_{i\leq n} i2^î +(n+1)2^{n+1}= [(n-1)2^{n+1}+2]+(n+1)2^{n+1}$ by induction hypothesis. Then subtracting the $i\leq n$ sum from both sides you are done.