Hello, Universe!

2018/12/30

I’m refurbishing my blog site anew, and this post doesn’t serve much purpose beyond driving that.

Perhaps we can find something fun to discuss.. 💡

Did you know that every square number can be represented as the sum of consecutive odd numbers?

$$ \begin{align} 1 & = 1 \\ 4 & = 1 + 3 \\ 9 & = 1 + 3 + 5 + 7 \\ 16 & = 1 + 3 + 5 + 7 + 9 \\ \end{align} $$

We can show it by induction: Assume $n^2 = S(n) = \sum_{k=0}^{n-1} 2k+1$. Then,

$$S(n+1) = n^2 + (2n + 1) = (n + 1)^2$$

Voilà! And we can also see it visually:

Each new layer is just a copy of the previous, but with two more dots added on. And we begin with just one.

Going further we can construct similar reasoning for triangles, pentagons, and other regular polygons. In the triangle below each new row is a copy of the previous, with one more dot added on.

Instead of sums of consecutive odd numbers, we can generalize to sums of an arithmetic series with different step-sizes:

$$\text{triangles} = \sum_{k=0}^{n-1} 1k+1$$ $$\text{squares} = \sum_{k=0}^{n-1} 2k+1$$ $$\text{pentagons} = \sum_{k=0}^{n-1} 3k+1$$

And in general for any $s$-sided regular polygon,

$$\text{s-sided} = \sum_{k=0}^{n-1} (s-2)k+1$$

These are called figurate numbers. See also:

Super interesting, right?