1+1+2分之1+1+2+3分之1+1+2+3+4分之1…………+1+2+3+4…+100分之1 怎么算额?

题目

1+1+2分之1+1+2+3分之1+1+2+3+4分之1…………+1+2+3+4…+100分之1 怎么算额?

答案

1+2+3+4+.+n
=(1+n)*n/2
1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+.+1/(12+3+.+100)
=1+1/[(1*2)/2]+1/[(2*3)/2]+1/[(3*4)/2]+1/[(4*5)/2]+.+1/[(100*101)/2]
=1+2/(1*2)+2/(2*3)+2/(3*4)+2/(4*5)+.+2/(100*101)
=1+2[1/(1*2)+1/(2*3)+1/(3*4)+1/(4*5)+.+1/(100*101)]
=1+2*(1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+.+1/100-1/101)
=1+2*(1-1/101)
=1+2*100/101
=1+200/101
=301/101