value of summation of $2^i\cdot i$ I'm trying to calculate the value of 
$$2^0\cdot0 + 2^1\cdot1 + 2^2\cdot2 + .... 2^n\cdot n$$
I figured this would be summation $2^i \cdot i$ from $i = 0$ to $n$. But iI'm unable to calculate its value.  
I have tried searching online but haven't been able to find a formula or any property that 
could simplify it (maybe there was which I might not have understood)
P.S.This is not a homework question, I need this value to prove a theorem.

WolframAlpha claims that
$$
\sum_{n=0}^{N}2^{n}n=2+2^{N+1}(N-1)
$$
We can verify this using induction. Clearly it holds for $N=0$. Now assuming the formula holds for $N$, we have
$$
\sum_{n=0}^{N+1}2^{n}n=2+2^{N+1}(N-1)+2^{N+1}(N+1)=2+2^{n+1}(2*N)=2+2^{N+2}(N+1-1)
$$
And so the formula holds for all $N\in\{0,1,2,\ldots\}$.
