A) \[\frac{1}{3}(\sqrt{3n+2}-\sqrt{2})\]
B) \[\sqrt{3n+2}-\sqrt{2}\]
C) \[\sqrt{3n+2}+\sqrt{2}\]
D) \[\frac{1}{3}(\sqrt{2}-\sqrt{3m+2})\]
E) \[\frac{1}{3}(\sqrt{3n+2}+\sqrt{2})\]
Correct Answer: A
Solution :
Let\[{{S}_{n}}=\frac{1}{\sqrt{2}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{11}}\] \[+.....+\frac{1}{\sqrt{3n-1}+\sqrt{3n+2}}\] \[=\frac{\sqrt{2}-\sqrt{5}}{-3}+\frac{\sqrt{5}-\sqrt{8}}{-3}+\frac{\sqrt{8}-\sqrt{11}}{-3}\]\[+...+\frac{\sqrt{3n-1}-\sqrt{3n+2}}{-3}\] \[=-\frac{1}{3}(\sqrt{2}-\sqrt{3n+2})\] \[=\frac{1}{3}(\sqrt{3n+2}-\sqrt{2})\]
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