jmc

We can list the equations $$\begin{array}{rl}f(2)& =1-2f(1),\\ f(3)& =-2-2f(2),\\ f(4)& =3-2f(3),\\ f(5)& =-4-2f(4),\\ & \dots ,\\ f(1985)& =-1984-2f(1984),\\ f(1986)& =1985-2f(1985).\end{array}$$Adding these equations, we get $$f(2)+f(3)+\cdots +f(1986)=(1-2+3-4+\cdots +1983-1984+1985)-2f(1)-2f(2)-\cdots -2f(1985).$$To find $1-2+3-4+\cdots +1983-1984+1985,$ we can pair the terms $$\begin{array}{rl}1-2+3-4+\cdots +1983-1984+1985& =(1-2)+(3-4)+\cdots +(1983-1984)+1985\\ & =(-1)+(-1)+\cdots +(-1)+1985\\ & =-\frac{1984}{2}+1985\\ & =993.\end{array}$$Hence, $$f(2)+f(3)+\cdots +f(1986)=993-2f(1)-2f(2)-\cdots -2f(1985).$$Then $$2f(1)+3f(2)+3f(3)+\cdots +3f(1985)+f(1986)=993.$$Since $f(1986)=f(1),$ $$3f(1)+3f(2)+3f(3)+\cdots +3f(1985)=993.$$Therefore, $f(1)+f(2)+f(3)+\cdots +f(1985)=\boxed{331}.$