The genius of Gauss

One of the most famous stories in mathematics is related to Gauss when he was a little boy.  It went as follows:

"It was said that to punish the kids for misbehavior, a teacher asked all students in the class to find the sum of all the natural numbers from 1 to 100.  Gauss got the answer quickly and correctly."

Ordinary folks like myself will take a while to figure this out.  They would likely add up the numbers two at a time, something like , then , then , and so forth.  As you can imagine, this can take a little while.

Now, it seems like Gauss already knew multiplication and division, and was able to come up with the following clever solution to the problem:

Let

Writing the terms of the sum backwards, we also have:

Adding these two equations "vertically" (i.e. adding the -th term in the sum of the first equation to the -th term in the sum of the second equation) gives:

It is now a simple calculation to see that

Of course, this can easily be generalized to give the following well-known formula for the sum of the first integers:

The genius of Gauss is not that he has learned how to multiply and divide ahead of his class, but that he is able to devise such a clever and simple solution to this problem using these tools.  Not many of us, including myself, can think of that even if we know multiplication and division.