1/(x+a)(x+a+1)=1/(x+a)-1/(x+a+1)
1/(x+1)(x+2)+1/(x+2)(x+3)...+1/(x+99)(x+100)+1/(x+100)
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+……+1/(x+99)-1/(x+100)+1/(x+100)
=1/(x+1)=1999/2000
x=1/1999