怎么解答这个题:1+一加二分之一再加一加二加三分之一再依次往下加，最后是一加二加三加四一直加到100

Question

一定要帮我解出来

Answer 1

1+1/(1+2)+1/(1+2+3)+...+1/(1+2+3+...100)=?

1/(1+2+3+...n)=1/[n(1+n)/2]=2/n(n+1)=2(1/n-1/(n+1))

所以原式=2(1-1/2)+2(1/2-1/3)+2(1/3-1/4)+...+2(1/100-1/101)

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

=2(1-1/101)

=200/101

Answer 2

1+1/[1+2]+1/[1+2+3]+...+1/[1+2+3..+100]
=1+2×[1/2-1/3]+2×[1/3-1/4]+...+2×[1/100-1/101]
=1+2×[1/2-1/3+1/3-1/4+...+1/100-1/101]
=1+2×[1/2-1/101]
=1+1-2/101
=2-2/101
=200/101