Prove using mathematical induction $1\cdot2+2\cdot2^2+3\cdot2^3+\ldots+n\cdot2^n=2[1+(n-1)2^n]$ 
Prove the result using Mathematical Induction $$1\cdot2+2\cdot2^2+3\cdot2^3+\ldots\ldots+n\cdot2^n=2[1+(n-1)2^n].$$

I've been stuck on this problem for hours, I have no idea how do even calculate it. The exponents throw me off. If anyone can help me break it down step-by-step, I would truly appreciate it.
Here's my attempt


You've made a mistake while doubling $2[1+(K-1)2^K]$.
To give a clearer and more concise presentation of the inductive step, I suggest writing one side at one row.
\begin{align}
& 2[1+(K-1)2^K] + (K+1) 2^{K+1} \\
&= 2 + (K-1) 2^{K+1} + (K+1) 2^{K+1} \\
&= 2 + 2K 2^{K+1} \\
&= 2[1 + K 2^{K+1}]
\end{align}
