Tính giá trị biểu thức \(A = \dfrac{1}{{1 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 5 }} + \dfrac{1}{{\sqrt 5 + \sqrt 7 }} \)\(+ ... + \dfrac{1}{{\sqrt {2019} + \sqrt {2021} }}\)
Trả lời bởi giáo viên
Ta có:
\(A = \dfrac{1}{{1 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 5 }} + \dfrac{1}{{\sqrt 5 + \sqrt 7 }} \)\(+ ... + \dfrac{1}{{\sqrt {2019} + \sqrt {2021} }}\)\(= \dfrac{{\sqrt 3 - 1}}{{\left( {1 + \sqrt 3 } \right)\left( {\sqrt 3 - 1} \right)}} \)\(+ \dfrac{{\sqrt 5 - \sqrt 3 }}{{\left( {\sqrt 3 + \sqrt 5 } \right)\left( {\sqrt 5 - \sqrt 3 } \right)}} \)\(+ ....... + \dfrac{{\sqrt {2021} - \sqrt {2019} }}{{\left( {\sqrt {2019} + \sqrt {2021} } \right)\left( {\sqrt {2021} - \sqrt {2019} } \right)}}\)\( = \dfrac{{\sqrt 3 - 1}}{{3 - 1}} + \dfrac{{\sqrt 5 - \sqrt 3 }}{{5 - 3}} \)\(+ ....... + \dfrac{{\sqrt {2021} - \sqrt {2019} }}{{2021 - 2019}}\)\( = \dfrac{{\sqrt 3 - 1}}{2} + \dfrac{{\sqrt 5 - \sqrt 3 }}{2} \)\(+ ...... + \dfrac{{\sqrt {2021} - \sqrt {2019} }}{2}\)\( = \dfrac{{\sqrt 3 - 1 + \sqrt 5 - \sqrt 3 + ....... + \sqrt {2021} - \sqrt {2019} }}{2}\)\( = \dfrac{{\sqrt {2021} - 1}}{2}\)
Hướng dẫn giải:
- Áp dụng: \(\dfrac{1}{{\sqrt a + \sqrt b }} = \dfrac{{\sqrt a - \sqrt b }}{{a - b}}\) với \(a , b>0\)