Question

If $\prod _{p=1}^r{e}^{ip\theta }=1$ where $\prod$ denotes the continued product, then the most general value of $\theta$ is

• A. $\frac{2n\pi }{r(r-1)},n\in Z$
• B. $\frac{2n\pi }{r(r+1)}n\in Z$
• C. $\frac{4n\pi }{r(r-1)},n\in Z$
• D. $\frac{4n\pi }{r(r+1)},n\in Z$

Answer 1

The correct option is D $\frac{4n\pi }{r(r+1)},n\in Z$

Given, $\prod _{p=1}^r{e}^{ip\theta }=1$
Hence, $e^{i\theta }\cdot e^{2i\theta }\cdots e^{ri\theta }=1$
$\Rightarrow e^{\frac{ir(r+1)}{2}\theta }=e^{i2n\pi },n\in Z$
$\Rightarrow \theta =\frac{4n\pi }{r(r+1)},n\in Z$