[tex]( + 1) + ( - 2) + ( + 3) + ( - 4) + ( + 5) + ( - 6) + ... + ( - 2018)[/tex]
[tex] = 1 - 2 + 3 - 4 + 5 - 6 + ... + 2017 - 2018[/tex]
[tex] = 1 + 3 + 5 + ... + 2017 - 2 - 4 - 6 -8- ... - 2018[/tex]
[tex]1+3+5+...+2017[/tex]
[tex]1+3+5+...+2n-1={n}^{2}[/tex]
[tex]2n-1=2017[/tex]
[tex]2n=2017+1[/tex]
[tex]2n=2018\:|\:\div2[/tex]
[tex]n=1009[/tex]
[tex]{n}^{2}={1009}^{2}[/tex]
[tex] = {1009}^{2} - 2(1 + 2 + 3 + 4+... + 1009)[/tex]
[tex] = {1009}^{2} - 2 \times \frac{1009(1009 + 1)}{2} [/tex]
[tex] = {1009}^{2}- 1009 \times 1010[/tex]
[tex] = 1009(1009 - 1010)[/tex]
[tex] = 1009 \times (-1)[/tex]
[tex] = -1009[/tex]
