Question

Please tell the answer

Answer 1

solve:
$\frac{1}{1 \sqrt{2} + 2 \sqrt{1} } + \frac{1}{2 \sqrt{3} + 3 \sqrt{2} } + \frac{1}{3 \sqrt{4} + 4 \sqrt{3} } + ..... + \frac{1}{398 \sqrt{399} + 399 \sqrt{398} } + \frac{1}{399 \sqrt{400} + 400 \sqrt{399} }$

first rationalizing each term, we get

$= ( \frac{1}{1 \sqrt{2} + 2 \sqrt{1} } \times \frac{ \sqrt{2} - 2 \sqrt{1} }{ \sqrt{2} - 2 \sqrt{1} } ) + (\frac{1}{2 \sqrt{3} + 3 \sqrt{2} } \times \frac{2 \sqrt{3} - 3 \sqrt{2} }{2 \sqrt{3} - 3 \sqrt{2} } )+ ( \frac{1}{3 \sqrt{4} + 4 \sqrt{3} } \times \frac{3 \sqrt{4} - 4 \sqrt{3} }{3 \sqrt{4} - 4 \sqrt{3} }) ...... + (\frac{1}{398 \sqrt{399} + 399 \sqrt{398} } \times \frac{398 \sqrt{399 } - 399 \sqrt{398} }{398 \sqrt{399} - 399 \sqrt{398} } ) + (\frac{1}{399 \sqrt{400} + 400 \sqrt{399} } \times \frac{399 \sqrt{400} - 400 \sqrt{399} }{399 \sqrt{400} - 400 \sqrt{399} } )$

$= ( \frac{ \sqrt{2} - 2 \sqrt{1} }{ 2 - 4} ) + (\frac{2 \sqrt{3} - 3 \sqrt{2} }{(4 \times 3) - (9 \times 2)} )+ ( \frac{3 \sqrt{4} - 4 \sqrt{3} }{(9 \times 4) - (16 \times 3) }) + ...... + ( \frac{398 \sqrt{399 } - 399 \sqrt{398} }{( {398}^{2} \times 399) - ( {399}^{2} \times 398)} ) + (\frac{399 \sqrt{400} - 400 \sqrt{399}}{( {399 \times }^{2} 400)- ( {400}^{2} \times 399) } )$

$= ( \frac{ \sqrt{2} - 2 \sqrt{1} }{ - 2} ) + (\frac{2 \sqrt{3} - 3 \sqrt{2} }{12 - 18} )+ ( \frac{3 \sqrt{4} - 4 \sqrt{3} }{36 - 48 }) + ...... + ( \frac{398 \sqrt{399 } - 399 \sqrt{398} }{63203196 - 63361998} ) + (\frac{399 \sqrt{400} - 400 \sqrt{399}}{63680400 - 63840000} )$

$= ( \frac{ \sqrt{2} - 2 \sqrt{1} }{ - 2} ) + (\frac{2 \sqrt{3} - 3 \sqrt{2} }{ - 6} )+ ( \frac{3 \sqrt{4} - 4 \sqrt{3} }{ - 12}) + ...... + ( \frac{398 \sqrt{399 } - 399 \sqrt{398} }{ - 158802} ) + (\frac{399 \sqrt{400} - 400 \sqrt{399}}{ - 159600} )$

$= - \frac{ \sqrt{2} }{2} + \frac{2}{2} - \frac{2 \sqrt{3} }{6} + \frac{3 \sqrt{6} }{6} - \frac{3 \sqrt{4} }{12} + \frac{4 \sqrt{3} }{12} + .... - \frac{398 \sqrt{399} }{158802} + \frac{399 \sqrt{398} }{158802} - \frac{399 \sqrt{400} }{159600} + \frac{400 \sqrt{399} }{159600}$

$= 1 + ( \frac{3 \sqrt{2} }{6} - \frac{ \sqrt{2} }{2} ) + ( \frac{4 \sqrt{3} }{12} - \frac{2 \sqrt{3} }{6} ) + ..... + ( \frac{400 \sqrt{399} }{159600} - \frac{398 \sqrt{399} }{158802} ) - \frac{399 \sqrt{400} }{159600}$

$= 1 + ( \frac{\sqrt{2} }{2} - \frac{ \sqrt{2} }{2} ) + ( \frac{ \sqrt{3} }{3} - \frac{ \sqrt{3} }{3} ) + ..... + ( \frac{\sqrt{399} }{399} - \frac{ \sqrt{399} }{399} ) - \frac{ 20}{400}$

$= 1 - \frac{20}{400}$

$= 1 - \frac{1}{20}$

$= \frac{20 - 1}{20}$

$= \frac{19}{20}$

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