# What is the sum of all odd numbers between 0 and 100?

##### 1 Answer

First, notice an interesting pattern here:

#1, 4, 9, 16, 25, ...#

The differences between perfect squares (starting at

#1, 3, 5, 7, 9, ...#

The sum of

Let's take another example. You can quickly prove that:

#1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100#

There are

Therefore, the sum of

#((99+1)/2)^2 = color(blue)(2500)#

Formally, you can write this as:

#color(green)(sum_(n=1)^N (2n-1) = 1 + 3 + 5 + ... + (2N - 1) = ((N+1)/2)^2)#

where