Graphical Proof of the Sum of the First N Integers

The blue squares in each row represent the numbers from 1 to 5.

The blue squares in each row represent the numbers from 1 to 5.

Obviously someone must have done this before, but it just occurred to me, so I thought I’d put it up.

The formula for the sum of the first N integers is (N^2 + N) / 2. For example, 1 + 2 + 3 + 4 + 5 = (5^2 + 5) / 2 = (25 + 5) / 2 = 30 / 2 = 15.

Here’s a graphic showing a 5 x 5 square. I’ve marked out squares representing the numbers from 1 to 5, in the rows from top to bottom. The area of the square, of course, is 5 x 5 = 25. The area marked in blue is the sum of the first 5 integers: 1 + 2 + 3 + 4 + 5.

The red section is half the square.

The red section is half the square.

To the right is the same graphic, with half the square marked off in red. Since the area of the square is 25, the area of the red section is half that, or 12.5.

The remaining blue sections are each half of one of the small squares. There are of course 5 of them, one for each row/column.

So the area of the red triangle is 5 x 5 / 2, and the area of the blue triangles is 5 x 1/2. Adding those together gives (5 x 5 + 5) / 2. Generalizing that, if the sides of the square are N, then the sum of the red area is N x N / 2,  and the blue areas is N / 2. Adding those together gives  (N x N + N) / 2, or (N^2 + N) / 2.

As I said, I’m sure this isn’t original, but I hadn’t seen it before, and I was pretty pleased to think of it.

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