Question

please solve this problem

Answer 1

Given;

$\frac{1}{{1 + \sqrt{2} } } + \frac{1}{ \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } \\ \\ this \: can \: also \: be \: written \: as \: \\ \\ \frac{1}{{ \sqrt{2} + 1 } } + \frac{1}{ \sqrt{3} + \sqrt{2} } + \frac{1}{ \sqrt{4} + \sqrt{3} } \\ \\ then \: on \: rationalising \: them : \\ \\ \frac{ \sqrt{2} - 1 }{ { \sqrt{2} }^{2} - {1}^{2} } + \frac{ \sqrt{3} - \sqrt{2} }{ { \sqrt{3} }^{2} - \sqrt{2}^{2} } + \frac{ \sqrt{4} - \sqrt{3} }{ { \sqrt{4} }^{2} - \sqrt {3}^{2} } \\ \\ \frac{ \sqrt{2} - 1}{1} + \frac{ \sqrt{3} - \sqrt{2} }{1} + \frac{ \sqrt{4} - \sqrt{3} }{1} \\ \\ \frac{ \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} }{1} \\ \\ \frac{ - 1 + \sqrt{4} }{1} \\ \\ \frac{ - 1 + 2}{1} \implies \: \frac{1}{1} = 1$

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