A) 1
B) \[-1\]
C) \[-\,i\]
D) \[\,i\]
Correct Answer: C
Solution :
Since, \[{{x}_{f}}=\cos \left( \frac{\pi }{{{3}^{r}}} \right)-i\sin \left( \frac{\pi }{3} \right)\] \[{{x}_{1}}.{{x}_{2}}.{{x}_{3}}...\infty =\cos \left( \frac{\pi }{{{3}^{1}}}+\frac{\pi }{{{3}^{2}}}+\frac{\pi }{{{3}^{3}}}+...\infty \right)\] \[-i\sin \left( \frac{\pi }{{{3}^{1}}}+\frac{\pi }{{{3}^{2}}}+\frac{\pi }{{{3}^{3}}}+...\infty \right)\] \[=\cos \left( \frac{\frac{\pi }{3}}{1-\frac{1}{3}} \right)-i\sin \left( \frac{\frac{\pi }{3}}{1-\frac{1}{3}} \right)\] \[=\cos \left( \frac{\pi }{2} \right)-i\sin \left( \frac{\pi }{2} \right)=-i\]
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