Question

If \( \omega \) is the complex cube root of unity, then the value of
\( \omega + \omega \left( \dfrac{1}{2} + \dfrac{3}{8} + \dfrac{9}{32} + \dfrac{27}{128} + \cdots \right) \) is

• A. \(-1\)
• B. \(1\)
• C. \(-i\)
• D. i

Hint:

Always use identities of roots of unity to simplify expressions.

Answer 1

The Correct Option is A

The series is geometric with:
\( a = \dfrac{1}{2}, \; r = \dfrac{3}{4} \)
Sum \( = \dfrac{a}{1 - r} = 2 \)
\( \Rightarrow \omega + 2\omega = 3\omega \)
Using property of cube roots of unity:
\( 1 + \omega + \omega^2 = 0 \Rightarrow 3\omega = -1 \)