\(S_1=2-4-6+8+10-12-14+16+18+...+1994-1996-1998+2000\)

\(=\left(2-4-6+8\right)+\left(10-12-14+16\right)+...+\left(1994-1996-1998+2000\right)\)

\(=0+0+...+0=0\)

\(\left(1+\dfrac{x+1}{17}\right)+\left(1+\dfrac{x+2}{16}\right)=\left(1+\dfrac{x+3}{15}\right)+\left(1+\dfrac{x+4}{14}\right)\)

\(\dfrac{x+18}{17}+\dfrac{x+18}{16}=\dfrac{x+18}{15}+\dfrac{x+18}{14}\)

\(\dfrac{x+18}{17}+\dfrac{x+18}{16}-\dfrac{x+18}{15}-\dfrac{x+18}{14}=0\)

\(\left(x+18\right)\left(\dfrac{1}{17}+\dfrac{1}{16}-\dfrac{1}{15}-\dfrac{1}{14}\right)=0\)

Vì : \(\dfrac{1}{17}< \dfrac{1}{15};\dfrac{1}{16}< \dfrac{1}{14}\Rightarrow\dfrac{1}{17}+\dfrac{1}{16}-\dfrac{1}{15}-\dfrac{1}{14}< 0\)

\(\Rightarrow x+18=0\Rightarrow x=0-18=-18\)