First slide

Special Series in Sequences and Series

Question

The sum to $n$ terms of the series $\frac{1}{\sqrt{7} + \sqrt{10}} + \frac{1}{\sqrt{10} + \sqrt{13}} + \frac{1}{\sqrt{13} + \sqrt{16}} + \dots$ is

Moderate

A

$\frac{1}{3} (\sqrt{7 + 3 n} - \sqrt{7})$
B

$\frac{\sqrt{4 + 3 n - 2}}{3}$
C

$\frac{1}{3} (\sqrt{10 + 3 n} - \sqrt{10})$
D

none of these

Solution

We have
$t_{r} = \frac{1}{\sqrt{4 + 3 r} + \sqrt{7 + 3 r}} = \frac{\sqrt{7 + 3 r} - \sqrt{4 + 3 r}}{3}$
Thus, sum to n terms is equal to
$\begin{matrix} \frac{1}{3} [(\sqrt{10} - \sqrt{7}) + (\sqrt{13} - \sqrt{10}) + \dots + (\sqrt{7 + 3 n} - \sqrt{4 + 3 n})] \\ = \frac{1}{3} (\sqrt{7 + 3 n} - \sqrt{7}) \end{matrix}$

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