I was first introduced to pushup pyramids at a hockey camp that I attended while I was in junior high. I’ve encountered them occasionally, but fairly infrequently, since then. Even so, every time they come up, I manually add up how many pushups are included in the pyramid. If we’re going up to 10 pushups, here’s how I usually add it up in my head:

\[\begin{aligned} p(10) = & 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 \\ = & (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) + 10 + (9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) \\ = & 2 \cdot 45 + 10 \\ = & 100 \end{aligned}\]

Okay, so a pushup pyramid up to 10 pushups is 100 pushups total. I’m starting to think that this is just a way for coaches to hide how many pushups they’re making you do… It’s usually at this point during the pushup pyarmid that I stop thinking about math and remember to focus on doing pushups.

Summation Formula

Now that I’m revisiting the math behind these pushup pyramids (while not simultaneously trying to do one), I’d like to see if I can come up with a generic formula to figure out how many pushups are in the pyramid.

Representing “the number of pushups in a pyramid up to \(n\)” as \(p(n)\), let’s revisit that 10-pushup pyramid:

\[p(10) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1\]

If we instead use an \(n\)-pushup pyramid, \(p(n)\) can be represented as:

\[p(n) = 1 + 2 + ... + (n - 1) + n + (n - 1) + ... + 2 + 1\]

Let’s see if we can simplify that (using summation notation and the relevant sum formula)

\[\begin{aligned} p(n) & = 1 + 2 + ... + (n - 1) + n + (n - 1) + ... + 2 + 1 \\ & = \sum_{k = 1}^{n} k + \sum_{k = 1}^{n - 1} k \\ & = \frac{n \cdot (n + 1)}{2} + \frac{(n - 1) \cdot n}{2} \\ & = \frac{n^2 + n + n^2 - n}{2} \\ p(n) & = n^2 \end{aligned}\]

Conclusion

Turns out that there’s a very simple formula for determining how many pushups are in a pushup pyramid!

\[p(n) = n^2\]

Does this conclusion shed light on some heretofore undiscovered principle of math? No. Does it use fancy complicated math? No. Is it interesting? Meh. Even so, am I happy that I finally sat down and figured out a generic formula for determining how many pushups are in a pushup pyramid? Yes! It’s been bothering me for over a decade!