Дробь 1\1*2+1\2*3+1\3*4.....1\99*100

1/n(n+1) = 1/n - 1/(n+1)

1/1*2 + 1/2*3 +1/3*4 + ..... + 1/99*100 = (1/1 - 1/2) + (1/2-1/3) + (1/3-1/4) +... + (1/99 - 1/100) = 1 -1/2 + 1/2 - 1/3 + 1/3 + ..... - 1/99 + 1/99 - 1/100 = 1 - 1/100 = 99/100