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?
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