How do I evaluate $\sum _{ i=1 }^{ 2^n } (\frac{i}{2^n} - \frac{i-1}{2^n})(1-\frac{i-1}{2^n})$

Question

What does this sum equal ?

$\sum _{ i=1 }^{ 2^n } (\frac{i}{2^n} - \frac{i-1}{2^n})(1-\frac{i-1}{2^n})$

The answer I'm getting is

$-\frac{1}{2^{n+1}} - \frac{1}{2^{2n+1}} + \frac{1}{2^{n}}$ but I have a feeling this is wrong.

I would appreciate your help.

Answer 1

Actually the sum equals to $\sum _{ i=1 }^{ 2^n } \frac{1}{2^n}(1-\frac{i-1}{2^n})$=$\sum _{ i=1 }^{ 2^n } (\frac{1}{2^n}-\frac{i-1}{4^n})$=$\sum _{ i=1 }^{ 2^n } \frac{1}{2^n}-\sum_{i=1}^{{2}^{n}}\frac{i-1}{4^n}$ So it is really easy to solve this question.The answer should be $\cfrac{1+{2}^{n}}{{2}^{n+1}}$.