1/1x2x3+1/2x3x4+1/3x4x5+.1/98x99x100=

1/1x2x3+1/2x3x4+1/3x4x5+.1/98x99x100=
=1/2[1/1*2-1/2*3]+1/2[1/2*3-1/3*4]+1/2[1/3*4-1/4*5]+...+1/2[1/98*99-1/99*100]
=1/2[1/1*2-1/2*3+1/2*3-1/3*4+1/3*4-1/4*5+...+1/98*99-1/99*100]
=1/2[1/2-1/99*100]
=1/2*4949/9900
=4949/19800