It's not clear when to apply + and when to apply - in your series. Assume:
1+2+3-4-5-6+7+8+9-10-11-12+...+991+992+993-994-995-996+997+998+999-1000.
Now pair like colours:
[(1+999)+(2+998)+(3+997)]-[(4+996)+(5+995)+(6+994)]+[(7+993)+(8+992)+(9+991)]...-1000.
3000-3000+3000-3000...-1000.
Next, we need to find out how many triad groupings there are. 1-999 contains 333 groupings (an odd number), so we are going to have an unmatched grouping giving us a surplus of +3000 and the final term which is -1000. (The sum would appear to be 3000-1000=2000, because positive 3000 precedes negative 3000.) 332 groupings will sum to zero.
All this makes the initial assumption of the series structure. If this assumption is incorrect, then the rest of the logic will be faulty. However, I hope the method outlined here helps.