Lời giải
Ta có: \(\frac{1}{{\sqrt 1 + \sqrt 2 }} + \frac{1}{{\sqrt 2 + \sqrt 3 }} + ... + \frac{1}{{\sqrt {24} + \sqrt {25} }}\)
\( = \frac{{\left( {\sqrt 2 - \sqrt 1 } \right)\left( {\sqrt 2 + \sqrt 1 } \right)}}{{\sqrt 1 + \sqrt 2 }} + \frac{{\left( {\sqrt 3 - \sqrt 2 } \right)\left( {\sqrt 3 + \sqrt 2 } \right)}}{{\sqrt 2 + \sqrt 3 }} + ... + \frac{{\left( {\sqrt {25} - \sqrt {24} } \right)\left( {\sqrt {25} + \sqrt {24} } \right)}}{{\sqrt {24} + \sqrt {25} }}\)
\( = \sqrt 2 - \sqrt 1 + \sqrt 3 - \sqrt 2 + ... + \sqrt {25} - \sqrt {24} \)
\( = \sqrt {25} - 1 = 5 - 1 = 4\).