Apostol's proof for the method of exhaustion

I'm a high school student currently going through Apostol's calculus. I'm not that familiar with proofs, but I learned Calculus in school up to partial fractions, but we focused more on problems instead of the concept/proof, so please bear with me. I already understand most of the method of exhaustion, but there's one thing that keeps bothering me.

Apostol wants to prove that for every integer n>=1

$$1^2+2^2+...+n^2=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}$$

To do that, he starts with the formula: $$(k+1)^3=k^3+3k^2+3k+1$$

Now, my question is, why did he start with this formula? For instance, if I didn't know anything about this proof and I wanted to start from scratch, what sort of thought process do I go through to come up say "Okay, to prove this idenity we start with this formula."

• To your last point—proving things, in general, is hard. Oftentimes as you advance in your mathematics career, you might see proofs that have an unintuitive starting point because someone came up with a clever substitution and went with it. So there will be many times where you'll say, "how did they come up with that?" Commented Mar 29, 2020 at 10:30

A thing that suggests that you should be aiming at a cubic polynomial is the fact that $$1+2+\cdots+n$$ is a quadratic expression on $$n$$, and now you are just one degree above that.

On the other hand, if indeed you have$$1^2+2^2+\cdots+n^2=P(n)$$for some cubic polynomial $$P(x)$$ and you want to prove this by induction, it is conveniente to know $$P(n+1)-P(n)$$, since you know that$$\bigl(1^2+2^2+\cdots+(n+1)^2\bigr)-\bigl(1^2+2^2+\cdots+n^2\bigr)=(n+1)^2.$$So, you want to have$$P(n+1)-P(n)=(n+1)^2.$$

Notice that

$$(k+1)^3-k^3=3k^2+3k+1$$

1.This expression contains the term $$k^2$$ 2.We can sum both sides (except ofcourse $$\sum_{i=0}^n i^2$$
Note: We could have used $$(k+2)^3-k^3$$ or for finding $$\sum_{k=1}^n k^4$$ we can sum both sides of $$\frac{(k+1)^4+(k-1)^4}{2}=k^4+6k^2+1$$ . You can use this technique for finding many different summitions.