A duplicate question probably exists, but I couldn't find one. Please let me know if one is found and I'll delete this.
We know that:
$$\sum_{r=0}^n r=\frac12n(n+1)$$ and that: $$\sum_{r=0}^n r^3 =\frac14 n^2(n+1)^2$$ and these are easily proven by induction: $$\frac{n(n+1)}{2}+(n+1)=\frac{n^2+n+2n+2}{2}=\frac{n^2+3n+2}{2}=\frac12(n+1)(n+2)$$ and: $$\frac{n^2(n+1)^2}{4}+(n+1)^3=\frac{n^2(n+1)^2+4(n+1)^3}{4}$$ $$= \frac{(n+1)^2(n^2+4n+4)}{4}=\frac 14(n+1)^2(n+2)^2$$
We can see that, from the results obtained, $$\sum_{r=0}^n r^3 =\bigg(\sum_{r=0}^n r\bigg)^2$$
What I am curious to know is, is there a reason why this is the case, other than the numbers just showing this?
Thanks in advance.
