Question

1-1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+···+1/（1+2+3+···+100）

过程

Answer 1

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

= 1 - 1/((1+2)*2/2) + 1/((1+3)*3/2) + ... + 1/((1+100)*100/2))

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

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

= 2/3 + 2(1/3-1/101)

= 2/3 + 2/3 - 2/101

= 398/303

Answer 2

原式=2/1*2+2/2*3+......+2/100*101
=2*(1/1*2+1/2*3+......+1/100*101)
=2*100/101
=200/101

Answer 3

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

1-1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+···+1/（1+2+3+···+100）
=1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+···+1/（1+2+3+···+100）-2/3
=1+2*[(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...+1/(1/100-1/101)]-2/3
=1+2*(1/2-1/101)-2/3
=1+2*99/202-2/3
=1+99/101-2/3
=200/101-2/3
=398/303