# Geometric Series Equivalence

#### seal308

Hello,
Why is this:

Equivalent to this:

Thank you.

#### ChipB

MHF Helper
Since summations only work with integers, you can replace the log(n) parts with an integer, such as k. The write a proof by induction to show that

$$\displaystyle \sum _{i=0} ^{k-1} 2^i = 2^k-1$$

#### Idea

we expect $$\displaystyle \sum _{i=0}^{\text{Log} n-1} 2^i$$ to be an integer

but $$\displaystyle 2^{\text{Log} n}-1$$ is not necessarily an integer