Question

The sum to n terms of the series $\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\cdots$ is:

• A. $\sqrt{2n+1}$
• B. $\frac{1}{2}\sqrt{2n+1}$
• C. $\sqrt{2n-1}$
• D. $\frac{1}{2}\{\sqrt{2n+1}-1\}$

Answer 1

The correct option is D $\frac{1}{2}\{\sqrt{2n+1}-1\}$

$S_n=\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\cdots \frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}$
$=\frac{1}{2}\left[\right(\sqrt{3}-\sqrt{1})+(\sqrt{5}-\sqrt{3})+(\sqrt{7}-\sqrt{5})+\cdots +\cdots (\sqrt{2n+1}-\sqrt{2n-1}\left)\right]$
$=\frac{1}{2}(\sqrt{2n+1}-1)$