1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+.+1/(x+99)(x+100)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+.+1/(x+99)-1/(x+100)
=1/x-1/(x+100)
=100/[x(x+100)]