Question

因为1/1×2=1-1/2,1/2×3=1/2-1/3.1/19×20=1/19-1/20

Answer 1

因为1/(1×3)=(1-1/3)/2,1/(3×5)=(1/3-1/5)/2.1/(99×101)=(1/99-1/101)/2

所以1/（1×3）+1/（3×5）+.+1/（99×101）

=(1-1/3)/2+(1/3-1/5)/2+(1/5-1/7)/2+.+(1/99-1/101)/2

=[(1-1/3)+(1/3-1/5)+(1/5-1/7)+.+(1/99-1/101)]/2

=(1-1/101)/2

=50/101