[tex]\displaystyle 1+3+5+...+2011 = \\ \\ =1+2+3+4+5+...+2011-(2+4+6+...+2010)= \\ \\ = \frac{2011(2011+1)}{2}-2(1+2+3+...+1005)= \\ \\ = \frac{2011 \cdot 2012}{2} -2 \cdot \frac{1005(1005+1)}{2}= \frac{4046132}{2}-2 \cdot \frac{1005 \cdot 1006}{2} = \\ \\ =2023066- \not 2 \cdot \frac{1011030}{\not 2} =2023066-1011030= \\ \\ =1006(2011-1005)=1006 \cdot 1006=1006^2-p.p [/tex]
