## Triangular number equations via pictures

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 $T_n$ denotes the $n$th triangular number (that is, the number of dots in a triangular figure with $n$ dots in the bottom row, $(n-1)$ dots in the next row, and so on, and one dot on the top).

1. $3T_n + T_{n-1} = T_{2n}$
2. $3T_n + T_{n+1} = T_{2n+1}$
3. $(2k+1) T_n + T_{kn - 1} = T_{(k+1)n}$ (this one is a generalization of the first one)
4. $T_n T_k + T_{n-1} T_{k-1} = T_{nk}$

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?

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### 8 Responses to Triangular number equations via pictures

1. Jean-Philippe Burelle says:

Very fun activity to do with free time :)
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!

2. Dave Indelicato says:

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

3. 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.

4. quantac says:

Figuring out (4) made me really happy!

This blog is great.