A) \[\frac{1+i\sqrt{3}}{2}\]
B) \[\frac{-1+i\sqrt{3}}{2}\]
C) \[\frac{1-i\sqrt{3}}{2}\]
D) \[\frac{-1-i\sqrt{3}}{2}\]
Correct Answer: A
Solution :
\[{{x}_{1}}.\,{{x}_{2}}.\,{{x}_{3}}......\infty \] \[=\left[ \cos \left( \frac{\pi }{4} \right)+i\sin \left( \frac{\pi }{4} \right) \right]\]\[\left[ \cos \left( \frac{\pi }{{{4}^{2}}} \right)+i\sin \left( \frac{\pi }{{{4}^{2}}} \right) \right]\,\left[ \cos \left( \frac{\pi }{{{4}^{3}}} \right)+i\sin \left( \frac{\pi }{{{4}^{3}}} \right) \right]\,.....\infty \] \[=\cos \left( \frac{\pi }{4}+\frac{\pi }{{{4}^{2}}}+\frac{\pi }{{{4}^{3}}}+....\infty \right)+i\sin \left( \frac{\pi }{4}+\frac{\pi }{{{4}^{2}}}+\frac{\pi }{{{4}^{3}}}+.....\infty \right)\] = \[\cos \,\left( \frac{\pi /4}{1-1/4} \right)+i\sin \,\left( \frac{\pi /4}{1-1/4} \right)\] = \[\cos \left( \pi /3 \right)+i\sin \,(\pi /3)=\frac{1+\sqrt{3}i}{2}\].
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