Evaluate 1^{2} – 2^{2} + 3^{2} – 4^{2} + 5^{2} – 6^{2} + … + 199^{2}.

We have 1 + (3

^{2}– 2^{2}) + (5^{2}– 4^{2}) + (7^{2}– 6^{2}) + … + (199^{2}– 198^{2})= 1 + 5 + 9 + 13 + … + 397.

This is an arithmetic sequence with first term 1, difference 4, last term 397. There are 100 terms.

The sum is

*n*/2 · (first + last) = 100/2 · (1 + 397) = (50) · (398) = 19,900.