Question

heii va rog sa ma ajutati urgent cuacest exercitiu ofer 99 de puncte si coroana celui care imi raspunde corect: (2+4+6+...+2016)-(1+3+5...+2015)

Answer 1

$\displaystyle (2+4+6+...+2016)-(1+3+5+...+2015) \\ \\ 2+4+6+...+2016=2(1+2+3+...+1008)= \\ \\ =2 \times \frac{1008(1008+1)}{2} =2 \times \frac{1008 \times 1009}{2} =\not2 \times \frac{1017072}{\not2} =1017072$

$\displaystyle 1+3+5+...+2015= \\ \\ =1+2+3+4+5+...+2015-(2+4+6+...+2014)= \\ \\ = \frac{2015(2015+1)}{2} -2(1+2+3+...+1007)= \\ \\ = \frac{2015 \times 2016}{2} -2 \times \frac{1007(1007+1)}{2} = \frac{4062240}{2} -2 \times \frac{1007 \times 1008}{2} = \\ \\ =2031120- \not2 \times \frac{1015056}{\not2} =2031120-1015056=1016064$

$(2+4+6+....+2016)-(1+3+5+...+2015)= \\ \\ =1017072-1016064=\boxed{1008}$