The sum to n terms of the series 11+3+13+5+15+7+⋯

13+1+15+3+17+5+⋯ =3−12+5−32+7−52+⋯+2n+1−2n−12 =12[2n+1−1]

The sum to n terms of the series 11+3+13+5+15+7+⋯

13+1+15+3+17+5+⋯ =3−12+5−32+7−52+⋯+2n+1−2n−12 =12[2n+1−1]

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

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$\sqrt{2 n + 1}$

$\frac{1}{2} \sqrt{2 n + 1}$

$\sqrt{2 n + 1} - 1$

$\frac{1}{2} {\sqrt{2 n + 1} - 1}$

answer is D.

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Detailed Solution

$\frac{1}{\sqrt{3} + \sqrt{1}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \dots = \frac{\sqrt{3} - 1}{2} + \frac{\sqrt{5} - \sqrt{3}}{2} + \frac{\sqrt{7} - \sqrt{5}}{2} + \dots + \frac{\sqrt{2 n + 1} - \sqrt{2 n - 1}}{2} = \frac{1}{2} [\sqrt{2 n + 1} - 1]$