1/(1+2)+1/(1+2+3)+……+1/(1+2+3……+100)怎么算?请问是怎么算的?

问题描述:

1/(1+2)+1/(1+2+3)+……+1/(1+2+3……+100)怎么算?
请问是怎么算的?

1/(1 2) 1/(1 2 3) … 1/(1 2 3… 100)
=(1/2 1/3)*2 (1/3 1/4)*2 … (1/100/1/101)*2
=(1/2 1/3-1/3 1/4-1/4 … 1/100-1/101)*2
=(1/2-1/101)*2
=1-2/101
=99/101

=101

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