1/(1+2)+1/(1+2+3)+.+1/(1+2+3+.+100)
=2/(2*3)+2/(3*4)+.+2/(100*101)
=2(1/2-1/3+1/3-1/4+1/4-1/5+.+1/100-1/101)
=2*(1/2-1/101)
=99/101