First slide

Special Series in Sequences and Series

Question

If $a_{1}, a_{2}, \dots, a_{n}$ are in A.P., with common difference $d \neq 0$ then sum of the series $\sin d [cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + \dots + cosec a_{n - 1} cosec a_{n}]$ is

Moderate

A

$\sec a_{1} - \sec a_{n}$
B

$cosec a_{1} - cosec a_{n}$
C

$\cot a_{1} - \cot a_{n}$
D

$\tan a_{1} - \tan a_{n}$

Solution

$\sin d [cosec a_{1} cosec a_{2} + cosec a_{2} cosec a_{3} + \dots + cosec a_{n - 1} cosec a_{n}]$
$\begin{array}{l} = \frac{\sin (a_{2} - a_{1})}{\sin a_{1} \sin a_{2}} + \frac{\sin (a_{3} - a_{2})}{\sin a_{2} \sin a_{3}} + \dots + \frac{\sin (a_{n} - a_{n - 1})}{\sin a_{n - 1} \sin a_{n}} \\ = (\cot a_{1} - \cot a_{2}) + (\cot a_{2} - \cot a_{3}) \\ + \dots + (\cot a_{n - 1} - \cot a_{n}) \\ = \cot a_{1} - \cot a_{n} \end{array}$

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