Q. The sum to $n$ terms of the series $\frac{1}{\sqrt{1} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{5}} + \ ...$ is
A) $\sqrt{2n + 1}$
B) $\sqrt{2n + 1} - 1$
C) $\frac{1}{2}\sqrt{2n + 1}$
D) $\frac{1}{2}(\sqrt{2n + 1} - 1)$
Explanation
(d)
Let$S_{n} = \frac{1}{\sqrt{1} + \sqrt{3}} + \frac{1}{\sqrt{3 + \sqrt{5}}} + ...$
$= \frac{1}{2}\lbrack(\sqrt{3} - 1) + (\sqrt{5} - \sqrt{3)} + (\sqrt{7} - \sqrt{5})$
$+ ... + (\sqrt{2n + 1} - \sqrt{2n - 1})\ \rbrack$
$= \frac{1}{2}(\sqrt{2n + 1} - 1)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.