How to solve $\sum_{j=0}^{k-1}2^{-j}$?

Question

So, I'm trying to solve this sum :

$$\sum_{j=0}^{k-1}2^{-j}$$

How should I proceed when I have $\sum_i a^{-i}$ ?

Answer 1

Hint

Use the fact that

$$a^{-i}=\left(\frac 1a\right)^i.$$

Answer 2

You sum is an "incomplete" (in the sense that the index does not go up to infinity) geometric series. $$ \sum_{j=0}^{k-1} 2^{-j} = \sum_{j=0}^{k-1} (\frac{1}{2})^j $$ You can then use the formula.