Cancellation in big sums

 One prominent trick to find a huge sum is to look for something that cancels. We take the example of the following problem with this key idea in mind:

For positive integers $p$ and $q$, let $f(pq) = (pq+p+q)(p-q)$. Find $$f(10)+f(11)+\dots+f(98)+f(99)$$ (2017 CMC 12A #11)

We try to cancel stuff. For this purpose, we observe that $f(pq)=-f(qp)$. This is quite useful because, now, terms like $f(pq)$ and $f(qp)$ all cancel! So which terms are left? The ones in which $p=q$ and the ones with $q=0$. The ones with $p=q$ are $0$ anyway, so we don't have to worry about them. The ones with $q=0$ turn out to be $f(p0) = p^2$, so the resultant sum is just $1^2 + 2^2 + \dots + 9^2 = \boxed{285}$.