A) 1
B) ? 1
C) i
D) -i
Correct Answer: C
Solution :
We have, \[{{x}_{n}}=\cos \left( \frac{\pi }{{{3}^{n}}} \right)+i\sin \left( \frac{\pi }{{{3}^{n}}} \right)\] \[\therefore \]\[{{x}_{1}}.{{x}_{2}}.{{x}_{3}}...\] \[=\cos \left( \frac{\pi }{3}+\frac{\pi }{{{3}^{2}}}+... \right)+i\sin \left( \frac{\pi }{3}+\frac{\pi }{{{3}^{2}}}+... \right)\] \[=\cos \left( \frac{\pi /3}{1-\frac{1}{3}} \right)+i\sin \left( \frac{\pi /3}{1+\frac{1}{3}} \right)\] \[=\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}=0+i=i\]
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