1+99 = 100, 2+98=100 etc. So the sum of integers between 1 and 100 is 100*49 + 50. Is this what the author refers to? I believe he has misunderstood the point in that math problem.
> S(1..n) = (n^2 + n) / 2
This was, if I recall, the very first example of inductive proofs we did at my university.