1/(1*3)+1/(3*5)+……+1/(99*101)
=1/2*(1-1/3)+1/2*(1/3-1/5)+.+1/2*(1/99-1/101)
=1/2*(1-1/3+1/3-1/5+.+1/99-1/101)
=1/2*(1-1/101)
=1/2*100/101
=50/101