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  4. The sum of the first n terms of the series (1/√2+√5)+(1/√5+√8)+(1/√8+√11)+..... is

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Q. The sum of the first $n$ terms of the series $ \frac{1}{\sqrt{2}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{11}}+..... $ is

KEAMKEAM 2008Sequences and Series

Solution:

Let $ {{S}_{n}}=\frac{1}{\sqrt{2}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{11}} $ $ +.....+\frac{1}{\sqrt{3n-1}+\sqrt{3n+2}} $
$=\frac{\sqrt{2}-\sqrt{5}}{-3}+\frac{\sqrt{5}-\sqrt{8}}{-3}+\frac{\sqrt{8}-\sqrt{11}}{-3} $ $ +...+\frac{\sqrt{3n-1}-\sqrt{3n+2}}{-3} $
$=-\frac{1}{3}(\sqrt{2}-\sqrt{3n+2}) $
$=\frac{1}{3}(\sqrt{3n+2}-\sqrt{2}) $