Đáp án + Giải thích các bước giải:
`A=(1)/(\sqrt{1}+\sqrt{2})+(1)/(\sqrt{2}+\sqrt{3})+(1)/(\sqrt{3}+\sqrt{4})+.....+(1)/(\sqrt{n-1}+\sqrt{n})`
`=(\sqrt{1}-\sqrt{2})/((\sqrt{1}-\sqrt{2})(\sqrt{1}+\sqrt{2}))+(\sqrt{2}-\sqrt{3})/((\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3}))+(\sqrt{3}-\sqrt{4})/((\sqrt{3}-\sqrt{4})(\sqrt{3}+\sqrt{4}))+....+(\sqrt{n-1}-\sqrt{n})/((\sqrt{n-1}-\sqrt{n})(\sqrt{n-1)+\sqrt{n}))`
`=(\sqrt{1}-\sqrt{2})/(1-2)+(\sqrt{2}-\sqrt{3})/(2-3)+(\sqrt{3}-\sqrt{4})/(3-4)+....+(\sqrt{n-1}-\sqrt{n})/(n-1-n)`
`=(\sqrt{1}-\sqrt{2}+\sqrt{2}-\sqrt{3}+\sqrt{3}-\sqrt{4}+....+\sqrt{n-1}-\sqrt{n})/(-1)`
`=(\sqrt{1}-\sqrt{n})/(-1)`
`=\sqrt{n}-1`
