I was carrying my sleepy cranky four-year-old up a hiking trail yesterday, which gave my brain time to wander, and I noticed something.

Take the common pattern made with adding things to a triangle, where you create a new row by increasing the number of things in the row by 1 each time, like so:

X

XX

XXX

XXXX

XXXXX

XXXXXX

XXXXXXX

XXXXXXXX

and so on.

As you go down the triangle, your running total looks like this:

1

3

6

10

15

21

28

36

and so on.

With me so far?

So I noticed that between the first and second row, your running total is multiplied by 3 (1 x 3 = 3). Between the second and third rows, your running total is multiplied by 2 (3 x 2 = 6). Between the third and fourth it got tricky.

But here’s what I noticed:

1 x 3/1 = 3

2 x 4/2 = 6

6 x 5/3 = 10

10 x 6/4 = 15

15 x 7/5 = 21

21 x 8/6 = 28

28 x 9/7 = 36

And, presumably, so on.

In other words, if you add one to the numerator and denominator of the previous fraction and multiply your current running total by the resulting number, you’ll get the next running total.

This blew my mind, because at first glance, the two patterns ({1, 3, 6, 10, 15} and {3/1, 4/2, 5/3, 6/4, 7/5}) are completely unrelated–and the second pattern isn’t one I’ve ever seen elsewhere. But surely there’s a reason for it.

I’m decent at understanding math, but my vocabulary is pretty limited. Can anyone explain in very small words why this pattern works?