1/2*3+1/3*4+1/4*5+1/5*6+...+1/99*100

1/2*3+1/3*4+1/4*5+1/5*6+...+1/99*100
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+...........+1/99-1/100
=1/2-1/100
=50/100-1/100
=49/100