1-2+3-4+5-6+...+2007-2008=...

$1 - 2 + 3 - 4 + 5 - 6 + ... + 2007 - 2008 =\\\\ (1 + 3 + 5 + ... + 2007) - 2(1 + 2 + 3 + ... + 1004) =\\\\\\ \boxed{1 + 3 + ... + 2n - 1 = n^2}\\\\ \boxed{1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2}}\\\\\\ (1 + 3 + 5 + ... + (2*1004 - 1)) - 2(1 + 2 + 3 + ... + 1004) =\\\\ 1004^2 - 2(\frac{1004*1005}{2}) =\\\\ 1004^2 - 1004*1005 =\\\\ 1004*(1004 - 1005) = 1004*(-1) = -1004$