Question

If \(\omega\) is the complex cube root of unity, then the value of \[ \omega+\omega\!\left(\frac12+\frac38+\frac{9}{32}+\frac{27}{128}+\cdots\right) \] is

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

Hint:

Always use \(1+\omega+\omega^2=0\) for cube roots of unity.

Answer 1

The Correct Option is A

Step 1: The series is a G.P. with \(a=\frac12,\ r=\frac34\) \[ S_\infty=\frac{a}{1-r}=2 \]
Step 2: \[ \omega+\omega S_\infty=3\omega \]
Step 3: Using \(1+\omega+\omega^2=0\Rightarrow 3\omega=-1\)