Tính 1 / (căn bậc hai 1 + căn bậc hai 2) + 1 / (căn bậc hai 2 + căn bậc hai 3)
Tính \(\frac{1}{{\sqrt 1 + \sqrt 2 }} + \frac{1}{{\sqrt 2 + \sqrt 3 }} + ... + \frac{1}{{\sqrt {99} + \sqrt {100} }}\).
Ta có:
\(\frac{1}{{\sqrt 1 + \sqrt 2 }} + \frac{1}{{\sqrt 2 + \sqrt 3 }} + ... + \frac{1}{{\sqrt {99} + \sqrt {100} }}\)
\( = \frac{{\sqrt 1 - \sqrt 2 }}{{\left( {\sqrt 1 + \sqrt 2 } \right)\left( {\sqrt 1 - \sqrt 2 } \right)}} + \frac{{\sqrt 2 - \sqrt 3 }}{{\left( {\sqrt 2 + \sqrt 3 } \right)\left( {\sqrt 2 - \sqrt 3 } \right)}} + ... + \frac{{\sqrt {99} - \sqrt {100} }}{{\left( {\sqrt {99} + \sqrt {100} } \right)\left( {\sqrt {99} - \sqrt {100} } \right)}}\)
\( = \frac{{\sqrt 1 - \sqrt 2 }}{{ - 1}} + \frac{{\sqrt 2 - \sqrt 3 }}{{ - 1}} + ... + \frac{{\sqrt {99} - \sqrt {100} }}{{ - 1}}\)
\( = \sqrt 2 - \sqrt 1 + \sqrt 3 - \sqrt 2 + ... + \sqrt {100} - \sqrt {99} \)
\( = \sqrt {100} - \sqrt 1 = 10 - 1 = 9\).