Aptitude Question ID: 107146

The simplified form of
\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}} is :
[A]\sqrt{3}
[B]3-\sqrt{3}
[C]3\sqrt{3}
[D]5-\sqrt{3}

\mathbf{3-\sqrt{3}}
\frac{1}{\sqrt{3}+\sqrt{4}}
= \frac{1}{\sqrt{3}+\sqrt{4}}\times \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}}
= \frac{\sqrt{4}-\sqrt{3}}{4-3} = \sqrt{4}-\sqrt{3}
similarly,\ \frac{1}{\sqrt{4}+\sqrt{5}} = \sqrt{5}-\sqrt{4}\ ......\ so\ on.
∴ Expression =
= \sqrt{4}-\sqrt{3}+\sqrt{5}-\sqrt{4}+\sqrt{6}-\sqrt{5}+\sqrt{7}-\sqrt{6}+\sqrt{8}-\sqrt{7}+\sqrt{9}-\sqrt{8} = \sqrt{9}-\sqrt{3} = 3-\sqrt{3}
Hence option [B] is correct answer.

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