A) \[\sqrt{2}{{(\sqrt{2}+1)}^{2}}\]
B) \[{{(\sqrt{2}+1)}^{2}}\]
C) \[5\sqrt{2}\]
D) \[3\sqrt{2}+\sqrt{5}\]
Correct Answer: A
Solution :
\[\frac{\sqrt{2}+1}{\sqrt{2}-1},\frac{1}{\sqrt{2}(\sqrt{2}-1)},\frac{1}{2},......\] Common ratio of the series \[=\frac{1}{\sqrt{2}(\sqrt{2}+1)}\] \[\therefore \] sum \[=\frac{a}{1-r}={\left( \frac{\sqrt{2}+1}{\sqrt{2}-1} \right)}/{\left( 1-\frac{1}{\sqrt{2}(\sqrt{2}+1)} \right)}\;\] \[=\frac{(\sqrt{2}+1)}{(\sqrt{2}-1)}.\,\frac{\sqrt{2}\,(\sqrt{2}+1)}{(1+\sqrt{2})}\]\[=\sqrt{2}{{(\sqrt{2}+1)}^{2}}\].
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