CHÚC BẠN HỌC TỐT !!!!!!!!!!
Đáp án:
Giải thích các bước giải:
🏅
`A = 1/{1 + \sqrt{5}} + 1/{\sqrt{5} + \sqrt{9}} + ... + 1/{\sqrt{2009} + \sqrt{2013}}`
`= {1 - \sqrt{5}}/{(1 + \sqrt{5})(1 - \sqrt{5})} + {\sqrt{5} - \sqrt{9}}/{(\sqrt{5} + \sqrt{9})(\sqrt{5} - \sqrt{9})} + ... + {\sqrt{2009} - \sqrt{2013}}/{(\sqrt{2009} + \sqrt{2013})(\sqrt{2009} - \sqrt{2013})}`
`= {1 - \sqrt{5}}/{- 4} + {\sqrt{5} - \sqrt{9}}/{- 4} + ... + {\sqrt{2009} - \sqrt{2013}}/{- 4}`
`= {1 - \sqrt{5} + \sqrt{5} - \sqrt{9} + ... + \sqrt{2009} - \sqrt{2013}}/{- 4}`
`= {1 - \sqrt{2013}}/{- 4}`
`= {\sqrt{2013} - 1}/{4}`
❤
`B = 1/{1 + \sqrt{4}} + 1/{\sqrt{4} + \sqrt{7}} + ... + 1/{\sqrt{2017} + \sqrt{2020}}`
`= {1 - \sqrt{4}}/{(1 + \sqrt{4})(1 - \sqrt{4})} + {\sqrt{4} - \sqrt{7}}/{(\sqrt{4} + \sqrt{7})(\sqrt{4} - \sqrt{7})} + ... + {\sqrt{2017} - \sqrt{2020}}/{(\sqrt{2017} + \sqrt{2020})(\sqrt{2017} - \sqrt{2020})}`
`= {1 - \sqrt{4}}/{- 3} + {\sqrt{4} - \sqrt{7}}/{- 3} + ... + {\sqrt{2017} - \sqrt{2020}}/{- 3}`
`= {1 - \sqrt{4} + \sqrt{4} - \sqrt{7} + ... + \sqrt{2017} - \sqrt{2020}}/{- 3}`
`= {1 - \sqrt{2020}}/{- 3}`
`= {\sqrt{2020} - 1}/{3}`
🌟
`C = 1/{\sqrt{2} - \sqrt{3}} - 1/{\sqrt{3} - \sqrt{4}} + ... + 1/{\sqrt{100} - \sqrt{101}}`
`= {\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{100} + \sqrt{101}}/{(\sqrt{100} - \sqrt{101})(\sqrt{100} + \sqrt{101})}`
`= {\sqrt{2} + \sqrt{3}}/{- 1} - {\sqrt{3} +\sqrt{4}}/{- 1} + ... + {\sqrt{100} + \sqrt{101}}/{- 1}`
`= {\sqrt{2} + \sqrt{3} - \sqrt{3} - \sqrt{4} + ... + \sqrt{100} + \sqrt{101}}/{- 1}`
`= {\sqrt{2} + \sqrt{101}}/{- 1}`
`= - \sqrt{2} -\sqrt{101}`
