Q57 / 170 · Manipal (MET) 2015 · Mathematics
If $w$ is an imaginary cube root of unity, then the value of $(1+\omega)\left(1+\omega^2\right)\left(1+\omega^3\right)\left(1+\omega^4\right) \ldots\left(1+\omega^{3 n}\right)$
- A$2^{3 n}$
- B$2^{2 n}$
- Correct answerC$2^n$
- DNone of these
Solution
We have,
$\begin{aligned} & (1+\omega)\left(1+\omega^2\right)\left(1+\omega^3\right)\left(1+\omega^4\right) \ldots\left(1+\omega^{3 n}\right) \\ & =\left[\left\{(1+\omega)\left(1+\omega^2\right)\right\}\left\{\left(1+\omega^4\right)\left(1+\omega^5\right)\right\} \ldots\right. \\ & \left\{\left(1+\omega^{3 n-2}\right)\left(1+\omega^{3 n-1}\right)\right\} \times\left(1+\omega^3\right)\left(1+\omega^6\right) \\ & \left.\left(1+\omega^9\right) \ldots\left(1+\omega^{3 n}\right)\right]\end{aligned}$
$\begin{aligned} & =2^n\left[\left\{\left(-\omega^2\right)(-\omega)\right\}\left\{\left(-\omega^2\right)(-\infty)\right] \ldots\left\{\left(-\omega^2\right)(-\infty)\right\}\right] \\ & \Rightarrow \quad\left[(1+w)\left(1+w^2\right)\left\{(1+w)\left(1+w^2\right)\right\}\left\{(1+w)\left(1+w^2\right)\right\}\right. \\ & \left.\quad \quad \ldots\left\{(1+w)\left(1+w^2\right)\right\}\right] \times(1+1)(1+1) \ldots(1+1) \\ & =2^n\end{aligned}$
Hence, the correct answer is option: 'C'.