Question

1x$(-1)^{2} + 2x (-1)^{3} +3x (-1)^{4} +....+2008x (-1)^{2009}$ =?

Answer 1

1x(-1)²+2x(-1)³+3x(-1)⁴+...+2008x(-1)²⁰⁰⁹=
1x-2x+3x-4x+...+2007x-2008x=
=x(1-2+3-4+...+2007-2008)=
=x[(1+3+5+...+2007)-(2+4+6+...+2008)]=
=x(1008016-1009020)= - 1004x

1+3+5+...+2007=(2007+1)[(2007-1):2+1]:2=2008*1004:2=1008016
2+4+6+...+2008=(2008+2)[(2008-2):2+1]:2=2010*1004:2=1009020