Question

Please help me out.​

Answer 1

Answer is 11

Just do this to the each fraction

$\frac{1}{ \sqrt{100} - \sqrt{99} } = \frac{1}{ \sqrt{100} - \sqrt{99} } \times \frac{ \sqrt{100} + \sqrt{99} }{ \sqrt{100} + \sqrt{99} } \\ = \frac{ \sqrt{100} + \sqrt{99} }{( \sqrt{100} - \sqrt{99})( \sqrt{100} - \sqrt{99}) } \\ = \frac{ \sqrt{100} + \sqrt{99} }{ {( \sqrt{100}) }^{2} - {( \sqrt{99}) }^{2} } \\ = \frac{ \sqrt{100} + \sqrt{99} }{100 - 99} \\ = \sqrt{100} + \sqrt{99}$

Similarly if you the same for all fractions, you get

$\sqrt{100 } + \sqrt{99} - \sqrt{99} - \sqrt{98} + \sqrt{98} + \sqrt{97} - \sqrt{97} + ................ - \sqrt{2} + \sqrt{2} + \sqrt{1}$

All the terms from will cancel out except

$\sqrt{100} \: and \: \sqrt{1}$

So we get

$\sqrt{100} + \sqrt{1} = 10 + 1 = 11$