Since the Gauss trick only depends on the fact that we are summing numbers that are equally spaced, we can now try it on summing odd numbers.

First, for something that you can check easily by hand,

The sum of all of the odd numbers up to 5 ( that is, 1+3+5) is

The sum of all of the odd numbers up to 19 is

The sum of all of the odd numbers up to 21 is

The sum of all of the odd numbers up to 999 is

If you look at your answers carefully you might see something interesting. We will see more about this in future problems but try to guess what might be going on.