[tex]\displaystyle 1+3+5+...+2001= \\ \\ =1+2+3+4+5+....+2001-(2+4+6+....+2000)= \\ \\ = \frac{2001(2001+1)}{2} -2(1+2+3+...+1000)= \\ \\ = \frac{2001 \times 2002}{2} -2 \times \frac{1000(1000+1)}{2} = \frac{4006002}{2} -2 \times \frac{1000 \times 1001}{2} = \\ \\ =2003001- \not 2 \times \frac{1001000}{\not 2} =2003001-1001000=1002001[/tex]
