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
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?
Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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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!
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)
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. 🙂
Figuring out (4) made me really happy!
This blog is great.
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