1/(1*3) + 1/(3*5) + 1/(5*7) + .+ 1/(97*99) + 1/(99*101) = (1-1/3)/2 + (1/3 - 1/5)/2 + (1/5-1/7)/2 + .+ (1/97-1/99)/2 + (1/99-1/101)/2 = [(1-1/3) + (1/3 - 1/5) + (1/5-1/7) + .+ (1/97-1/99) + (1/99-1/101)] / 2 = [(1+1/3+1/5+.+1/97+1/99) - (1/3+1/5+1/7+.+1/99+1/101)] / 2 = [ 1 - 1/101] / 2 = [100/101] / 2 = 50/101