Finding a closed formula for $\sum_{i=0}^n 2^i \cdot (n-i)$ through the perturbation method

Question

I need to find a closed form for

$$\sum_{i=0}^n 2^i \cdot (n-i)$$

Through the perturbatino method.

How could I start?

May I reduce the summation in multiple simpler summations?

Answer 1

$$\sum_{i=0}^n2^i\cdot(n-i)=\sum_{i=0}^ni\cdot2^{n-i}=\sum_{i=1}^ni\cdot2^{n-i}=\sum_{i=0}^n\frac{i\cdot2^n}{2^i}=2^n\sum_{i=0}^n\frac{i}{2^i}=2^n\cdot2=2^{n+1}$$