[tex]\displaystyle 1+3+5+7+...+2013-(2+4+6+...+2012) \\ \\ 1+3+5+7+...+2013= \\ \\ =1+2+3+4+5+6+7+...+2013-(2+4+6+...+2012)= \\ \\ = \frac{2013(2013+1)}{2} -2(1+2+3+...+1006)= \\ \\ = \frac{2013 \times 1014}{2} -2 \times \frac{1006(1006+1)}{2} = \frac{4054182}{2} -2 \times \frac{1006 \times 1007}{2} = \\ \\ =2027091-\not2 \times \frac{1013042}{\not 2} =2027091-1013042=1014049 [/tex]
[tex]\displaystyle 2+4+6+...+2012=2(1+2+3+...+1006)= \\ \\ =2 \times \frac{1006(1006+1)}{2} =2 \times \frac{1006 \times 1007}{2} = \not 2 \times \frac{1013042}{\not 2} =1013042 \\ \\ 1+3+5+7+...+2013-(2+4+6+...+2012)= \\ \\ =1014049-1013042=\boxed{1007}[/tex]
