Ta có:$x_1=x_1,x_2=x_2$
$x_3=x_2-x_1$
$x_4=x_3-x_2=-x_1$
$x_5=x_4-x_3=-x_2$
$x_6=x_5-x_4=x_1-x_2$
$x_7=x_6-x_5=x_1$
$x_8=x_7-x_6=x_2$
Bắng quy nạp ta chứng minh được: $x_{6k+i}=x_i,S_{6k}=0$
Khi đó:
$S_{2003}=1987\Leftrightarrow S_{2004}-x_{2004}=1987 \Leftrightarrow x_{2004}=-1987 \Leftrightarrow x_1-x_2=-1987$
$S_{1987}=2003\Leftrightarrow S_{1986}+x_{1987}=2003\Leftrightarrow x_{1987}=2003\Leftrightarrow x_1=2003$
Từ đó suy ra: $x_2=3990$
$S_{2008}=S_4=2x_2-x_1=5977$