Gauss algorithm to calculate 1:2008 plus 1:1004 plus 3:2008. Add 2007:2008 plus 2008:2008

Gauss algorithm to calculate 1:2008 plus 1:1004 plus 3:2008. Add 2007:2008 plus 2008:2008

[(1+2008)*2008/2]/2008=1004.5

2007/2008+9×2007/2008+99×2007/2008+1/502

2007 / 2008 + 9 × 2007 / 2008 + 99 × 2007 / 2008 + 1 / 502 = (1 + 9 + 99) × 2007 / 2008 + 4 / 2008 = 109 * 2007 / 2008 + 4 / 2008 = 109 * (1-1 / 2008) + 4 / 2008 = 109-109 / 2008 + 4 / 2008 = 109-105 / 2008 = 108-1903 / 2008

Is the sum of 1 + (- 2) + 3 + (- 4) + 5 + (- 6) +. + 2007 + (- 2008) + 2009 - 1004 or 1005?

1005

(2007-1)+(2006-2)+(2005-3)+…… +(1005-1003)

(2007-1)+(2006-2)+(2005-3)+…… +(1005-1003)
=(2007-1003)+(2006-10022)+(2005-1001)+…… +(1005-1)
=1004×1003
=1007012