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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class11  >>  Sequence and Series
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Sum of series $\;S=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\;....\;+\frac{1}{100\sqrt{99}+99\sqrt{100}} \;is$

$(a)\;\frac{1}{10}\qquad(b)\;\frac{9}{10}\qquad(c)\;\frac{3}{10}\qquad(d)\;None$

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Answer : $(b)\;\frac{9}{10}$
Explanation : $a_{k}=\frac{1}{\sqrt{k+1}\;k+\sqrt{k}\;(k+1)}$
$=\frac{1}{\sqrt{k(k+1)}}\;[\frac{1}{\sqrt{k}+\sqrt{k+1}}]$
$\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k\;(k+1)}}$
$=\;[\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}]$
$S=\sum_{k=1}^{99}\;\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}$
$=1-\frac{1}{\sqrt{100}}$
$=\frac{9}{10}.$
answered Jan 20, 2014 by yamini.v
 

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