[tex]1\cdot (-1)^1+2\cdot (-1)^2+3\cdot (-1)^3+...+250\cdot (-1)^{250} =\\ \\ = 1\cdot (-1)+2\cdot 1+3\cdot (-1)+...+250\cdot 1 \\ \\ =-1+2-3+4-5+...-249+250\\ \\ = (2-1)+(4-3)+(6-5)+...+(250-249)\\ \\ =1+1+1+\underset{\text{de 125 ori}}{\underbrace{...}}+1\\ \\ = 1\cdot 125\\ \\ = \boxed{125}[/tex]
