How to Solve Summation by Hand I am having trouble solving the following summation by hand.
$$ \sum_{i=0}^{n-1} 2^i (n-i) $$
Can someone guide me in the right direction, especially for the $i \cdot 2^i$ part?

Alternatively, one can use generating function method to find the sum: We know that
$$ \sum_{n=0}^{\infty} 2^n x^n = \frac{1}{1-2x}
\qquad \text{and} \qquad
\sum_{n=0}^{\infty} n x^n = \frac{x}{(1-x)^2}. $$
Thus it follows that
\begin{align*}
\sum_{n=0}^{\infty} \bigg( \sum_{i=0}^{n} 2^i (n-i) \bigg) x^n
&= \frac{1}{1-2x} \cdot \frac{x}{(1-x)^2} \\
&= \frac{2}{1-2x} - \frac{1}{1-x} - \frac{1}{(1-x)^2} \\
&= \sum_{n=0}^{\infty} ( 2^{n+1} - 1 - (n+1) ) x^n.
\end{align*}
Therefore 
$$ \sum_{i=0}^{n} 2^i (n-i) = 2^{n+1} - n - 2. $$
