Question

what is the value of 1/ 1+root 2 +1/root 2+ root 3 + 1/root 3 + root 4 .... upto 15 terms

Answer 1

To FinD :

$\rightsquigarrow \sf \: \frac{1}{ \sqrt{1} + \sqrt{2} } + \frac{1}{ \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } + \dots + \frac{1}{ \sqrt{14} + \sqrt{15} } = {?}$

SolutioN :

$\: \: \: \: \: \: \: \: \: \: \: \sf \: \frac{1}{ \sqrt{1} + \sqrt{2} } + \frac{1}{ \sqrt{2} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{4} } + \dots + \frac{1}{ \sqrt{14} + \sqrt{15} } {}$

$\implies \sf \: \frac{ \sqrt{1} - \sqrt{2} }{( \sqrt{1} + \sqrt{2})( \sqrt{1} - \sqrt{2} ) } + \frac{ \sqrt{2} - \sqrt{3} }{ (\sqrt{2} + \sqrt{3})( \sqrt{2} - \sqrt{3}) } + \dots + \frac{ \sqrt{14} - \sqrt{15} }{( \sqrt{14} + \sqrt{15} )( \sqrt{14} - \sqrt{15}) }$

$\implies \sf \: \frac{ \sqrt{1} - \sqrt{2} }{ - 1} + \frac{ \sqrt{2} - \sqrt{3} }{ - 1} + \dots + \frac{ \sqrt{14} - \sqrt{15} }{ - 1}$

$\: \: \: \: \: \: \: \: \: \: \therefore{ \underline{ \boxed{\sf{ \sqrt{15} - 1}}}}$

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