Đáp án:
`1/(1+sqrt2)+1/(sqrt2+sqrt3)+1/(sqrt3+sqrt4)+....+1/(sqrt{99}+sqrt{100})`
`=(sqrt2-1)/((1+sqrt2)(sqrt2-1))+(sqrt3-sqrt2)/((sqrt3+sqrt2)(sqrt3-sqrt2))+(sqrt4-sqrt3)/((sqrt3+sqrt4)(sqrt4-sqrt3))+......+(sqrt{100}-sqrt{99})/((sqrt{99}+sqrt{100})(sqrt{100}-sqrt{99}))`
`=(sqrt2-1)/(2-1)+(sqrt3-sqrt2)/(3-2)+(sqrt4-sqrt3)/(4-3)+.......+(sqrt{100}-sqrt{99})/(100-99)`
`=sqrt2-1+sqrt3-sqrt2+sqrt4-sqrt3+......+sqrt{100}-sqrt{99}`
`=sqrt{100}-1`
`=10-1=9`