Question

1÷1+1÷3+1÷6+1÷10+1÷15+1÷21+…+1÷5050

Answer 1

整理为：

1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+…+1/(1+2+3+…+100)

1/(1+2+3+…+n)=2/n(n+1),又1/n(n+1)=1/n-1/(n+1)

所以原式化简为：

1+2/2*3+2/3*4+2/4*5+…2/100*101

=2(1/2+1/2*3+1/3*4+…1/100*101)

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

=2(1-1/101)

=200/101