First slide

Special Series in Sequences and Series

Question

$Find the sum of the series \sum_{k = 1}^{360} (\frac{1}{k \sqrt{k + 1} + (k + 1) \sqrt{k}})$

Moderate

A

$\frac{19}{18}$
B

$\frac{18}{19}$
C

$\frac{17}{18}$
D

$\frac{16}{18}$

Solution

$\begin{array}{l} T_{k} = \frac{1}{\sqrt{k} \sqrt{k + 1} (\sqrt{k} + \sqrt{k + 1})} \\ = \frac{\sqrt{k + 1} - \sqrt{k}}{\sqrt{k} \sqrt{k + 1}} \\ = \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k + 1}} \\ ∴ S = \sum_{k = 1}^{360} (\frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k + 1}}) \\ = 1 - \frac{1}{\sqrt{361}} \\ = 1 - \frac{1}{19} = \frac{18}{19} \end{array}$

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