Question

Ajutor!!!!!
2+4+6+...+2014=?
1+3+5+...+2013=?

Answer 1

$\displaystyle a).2+4+6+...+2014=2(1+2+3+...+1007)= \\ \\ =2 \times \frac{1007(1007+1)}{2} =2 \times \frac{1007 \times 1008}{2} =\not2 \times \frac{1015056}{\not2} =1015056$

$\displaystyle b).1+3+5+...+2013= \\ \\ =1+2+3+4+5+...+2013-(2+4+6+...+2012)= \\ \\ = \frac{2013(2013+1)}{2} -2(1+2+3+...+1006)= \\ \\ = \frac{2013 \times 2014}{2} -2 \times \frac{1006(1006+1)}{2} = \frac{4054182}{2} -2 \times \frac{1006 \times 1007}{2} = \\ \\ =2027091-\not2 \times \frac{1013042}{\not2} =2027091-1013042=1014049$