Question

Please Solve this question​

Answer 1

$= = > \: \: \frac{1}{1 + \sqrt{2} } + \frac{1}{ \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } + \frac{1}{ \sqrt{4} + \sqrt{5} } + \frac{1}{ \sqrt{5} + \sqrt{6} } + \frac{1}{ \sqrt{6} + \sqrt{7} } + \frac{1}{ \sqrt{7} + \sqrt{8} } + \frac{1}{ \sqrt{8} + \sqrt{9} } \\ \\ = = > \: \: \frac{1}{1 + \sqrt{2} } \times \frac{1 - \sqrt{2} }{1 - \sqrt{2} } + \frac{1}{ \sqrt{2} + \sqrt{3} } \times \frac{ \sqrt{2} - \sqrt{3} }{ \sqrt{2} - \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } \times \frac{ \sqrt{3} - \sqrt{4} }{ \sqrt{3} - \sqrt{4} } + \frac{1}{ \sqrt{4} + \sqrt{5} } \times \frac{ \sqrt{4} - \sqrt{5} }{ \sqrt{4} - \sqrt{5} } + \frac{1}{ \sqrt{5} + \sqrt{6} } \times \frac{ \sqrt{5} - \sqrt{6} }{ \sqrt{5} - \sqrt{6} } + \frac{1}{ \sqrt{6} + \sqrt{7} } \times \frac{ \sqrt{6} - \sqrt{7} }{ \sqrt{6} - \sqrt{7} } + \frac{1}{ \sqrt{7} + \sqrt{8} } \times \frac{ \sqrt{7} - \sqrt{8} }{ \sqrt{7} - \sqrt{8} } + \frac{1}{ \sqrt{8} + \sqrt{9} } \times \frac{ \sqrt{8} - \sqrt{9} }{ \sqrt{8} - \sqrt{9} } \\ \\ = = > \: \: \frac{1 - \sqrt{2} }{ {1}^{2} - {( \sqrt{2} )}^{2} } + \frac{ \sqrt{2} - \sqrt{3} }{ {( \sqrt{2} )}^{2} - {( \sqrt{3} )}^{2} } + \frac{ \sqrt{3} - \sqrt{4} }{ {( \sqrt{3} )}^{2} - {( \sqrt{4}) }^{2} } + \frac{ \sqrt{4} - \sqrt{5} }{ {( \sqrt{4} )}^{2} - {( \sqrt{5} )}^{2} } + \frac{ \sqrt{5} - \sqrt{6} }{ {( \sqrt{5}) }^{2} - {( \sqrt{6} )}^{2} } + \frac{ \sqrt{6} - \sqrt{7} }{ {( \sqrt{6}) }^{2} - {( \sqrt{7} )}^{2} } + \frac{ \sqrt{7} - \sqrt{8} }{ {( \sqrt{7}) }^{2} - {( \sqrt{8}) }^{2} } + \frac{ \sqrt{8} - \sqrt{9} }{ {( \sqrt{8}) }^{2} - {( \sqrt{9} )}^{2} } \\ \\ = = > \: \: - \frac{1 - \sqrt{2} }{1} - \frac{ \sqrt{2} - \sqrt{3} }{1} - \frac{ \sqrt{3} - \sqrt{4} }{1} - \frac{ \sqrt{4} - \sqrt{5} }{1} - \frac{ \sqrt{5} - \sqrt{6} }{1} - \frac{ \sqrt{6} - \sqrt{7} }{1} - \frac{ \sqrt{7} - \sqrt{8} }{1} - \frac{ \sqrt{8} - \sqrt{9} }{1} \\ \\ = = > \: \: - (1 - \sqrt{2} ) - ( \sqrt{2} - \sqrt{3} ) - ( \sqrt{3} - \sqrt{4} ) - ( \sqrt{4} - \sqrt{5} ) - ( \sqrt{ 5} - \sqrt{6} ) - ( \sqrt{6} - \sqrt{7} ) - ( \sqrt{7} - \sqrt{8} ) - ( \sqrt{8} - \sqrt{9} ) \\ \\ = = > \: \: -1 + \sqrt{2} - \sqrt{2} + \sqrt{3} - \sqrt{3} + \sqrt{4} - \sqrt{4} + \sqrt{5} - \sqrt{5} + \sqrt{6} - \sqrt{6} + \sqrt{7} - \sqrt{7} + \sqrt{8} - \sqrt{8} + \sqrt{9} - \sqrt{9} \\ \\ = = > \: \: -1$