Another Proof of the Sum of the First n Positive Integers

We have discussed how Gauss was able to devise  a clever way to add the first 100 positive integers at a very young age in a few minutes. We generalized his method and have also seen the link between the sum and the area of a triangle. In both discussions, we have shown that the sum of the first n positive integers is \frac{n(n+1)}{2}.

In this post, we discuss another geometric proof of the problem above.  We start with a specific case adding the first 6 positive integers, and the proceed to the general case.

We can add the first 6 positive integers manually or just by counting the number of unit squares in the figure above. The figure shows the geometric representation of the sum of the first 6 positive integers. Adding manually is not difficult if the given is small. However, we need a strategy for larger numbers. » Read more