First slide

Introduction to ITF

Question

If $n \in N, \sum_{k = 1}^{n} S i n^{- 1} (x_{k}) = \frac{n π}{2}$ then $\sum_{k = 1}^{n} (x_{k}) =$

Moderate

A

$n$
B

$k$
C

$\frac{k (k + 1)}{2}$
D

$\frac{n (n + 1)}{2}$

Solution

put, ${\sum \sin^{- 1}}_{k = 1}^{n} (x_{k}) = \frac{n π}{2} \Rightarrow \sin^{- 1} (x_{k}) = \frac{π}{2} f o r k = 1, 2, . . . . n \Rightarrow x_{k} = \sin \frac{π}{2} f o r k = 1, 2, . . . . n \Rightarrow x_{1} = x_{2} = x_{3} = ........... = 1$
$∴ \sum_{k = 1}^{n} x_{k} = 1 + 1 + 1 + ............ + 1 (n times) = n$

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