Let $S = 1 - 2 + 3 - 4 + \cdots + 2009 - 2010$. What is the residue of $S$, modulo 2010?

Using arithmetico-geometric series, S = -1004.  In mod 2010, this is 2010 - 1004 = 1006.

incorrect

∑[ (-1)^(n+1) * n, n, 1, 2010] =-1005

-1005 mod 2010 = - 1005

2010 + (- 1005) = 1005