The fourth triangular number

The other day I was fiddling around a bit with

triangular numbers. By

*only drawing pictures* I was able to come up with the following triangular number equations, where

denotes the

th triangular number (that is, the number of dots in a triangular figure with

dots in the bottom row,

dots in the next row, and so on, and one dot on the top).

- (this one is a generalization of the first one)

Now, none of these are hard to prove algebraically, but that’s not the point. Can you come up with *pictures* to illustrate the validity of each equation? Can you use a picture to figure out how to generalize #2 in the same way that #3 generalizes #1?

39.953605
-75.213937

##
About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

Very fun activity to do with free time ๐

I solved all of them without too much problems, but since I had solved #1 in the way depicted with the smarties, I had a little trouble thinking how to “generalize” it…

There seem to be multiple ways to draw those properties! I didn’t have too much trouble generalizing #2 either, I just “sank” some of the triangles in the picture for #3, haha.

Thank you for the nice problem suggestion!

I don’t know if this is the best way to express it, but it looks like #2 generalizes as:

(2k + 1)Tsub(n) + Tsub(k(n+1)) = Tsub(k(n+1) + n)

Fun activity.

Dave

This is a nice activity. What caught my attention are the M and M’s. I wanna use it as logo for my blog, Mathematics and Multimedia, but I am afraid, I might get sued. LOL.

Go ahead, you won’t get sued! I took that picture myself. See https://mathlesstraveled.com/license/ .

Thanks Brent. ๐

Figuring out (4) made me really happy!

This blog is great.

Pingback: Wild About Math bloggers 4/1/11 » Fun Math Blog

Pingback: Triangular number equations via pictures: solutions | The Math Less Traveled