how to solve the summation

Question

This question might have been asked before but I have not been able to find it.

How can I find: $$\sum_{i=0}^n 2^{x-i}$$

Help?

Answer 1

Use $$\sum_{k=0}^{n} x^{k} = \frac{1 - x^{n+1}}{1-x}$$ to obtain $$\sum_{k=0}^{n} \frac{1}{2^{k}} = 2 - \frac{1}{2^{n}}.$$ This leads to $$\sum_{k=0}^{n} \frac{2^{m}}{2^{k}} = 2^{m+1} - \frac{2^{m}}{2^{n}}.$$

If $m=n$ then $$\sum_{k=0}^{n} 2^{n-k} = 2^{n+1} - 1.$$