Conceptual basis for formula for sum of n first integers raised to power k, Most programmers (including me) are painfully aware of quadratic behavior resulting from a loop that internally performs 1, 2, 3, 4, 5 and so on operations per iteration,
$$\sum_{i=1}^n i = \frac{n \left(n+1\right)}{2} $$
It’s very easy to derive, e.g. considering the double of the sum like $(1+2+3) + (3+2+1) = 3\times4$ .
For the sum of squares there is a very similar formula,
$$\sum_{i=1}^n i^2 = \frac{n(n+\frac{1}{2})(n+1)}{3}$$
But the only way I found to derive that was to assume it would sum to a cubic polynomial $f(x)=Ax^3+Bx^2+Cx+D$, and solve for $f(n)-f(n-1)=n^2$.
I’m guessing that there is a much simpler system and concept here, generalizing to $\sum_{i=1}^n i^k$?

Your derivation is actually quite nice. I doubt you'll find some very conceptually simple way of computing the general formula (so that you could look at it and say: aha! that was why...!). There are some "visual proofs" for power one and two but I don't think they can be generalized.
Mathematically, for can look here and here. But this, I guess,  is not what you're looking for.
