Complex Numbers And Quadratic Equations question 226
Question: If $ x_{n}=\cos ( \frac{\pi }{4^{n}} )+i\sin ( \frac{\pi }{4^{n}} ), $ then $ x_1.x_2.x_3….\infty = $ [EAMCET 2002]
Options:
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} $
Show Answer
Answer:
Correct Answer: A
Solution:
$ x_1.x_2.x_3……\infty $ $ =[ \cos ( \frac{\pi }{4} )+i\sin ( \frac{\pi }{4} ) ] $ $ [ \cos ( \frac{\pi }{4^{2}} )+i\sin ( \frac{\pi }{4^{2}} ) ][ \cos ( \frac{\pi }{4^{3}} )+i\sin ( \frac{\pi }{4^{3}} ) ]…..\infty $ $ =\cos ( \frac{\pi }{4}+\frac{\pi }{4^{2}}+\frac{\pi }{4^{3}}+….\infty )+i\sin ( \frac{\pi }{4}+\frac{\pi }{4^{2}}+\frac{\pi }{4^{3}}+…..\infty ) $ = $ \cos ( \frac{\pi /4}{1-1/4} )+i\sin ( \frac{\pi /4}{1-1/4} ) $ = $ \cos ( \pi /3 )+i\sin (\pi /3)=\frac{1+\sqrt{3}i}{2} $ .
BETA
