If $z$ and $w$ are two non-zero complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$, then the value of $5iz + w$ is equal to
Step-by-Step Solution
Key Concept: The sum of all $n$-th roots of unity equals zero, allowing us to isolate the desired sum.
The sum $\sum_{k=1}^{4}\left(\sin\frac{2k\pi}{5} - i\cos\frac{2k\pi}{5}\right)$ can be rewritten using $\sin\theta - i\cos\theta = -i(\cos\theta + i\sin\theta) = -ie^{i\theta}$. This equals $-i\sum_{k=1}^{4}e^{i2k\pi/5}$. The sum $\sum_{k=1}^{4}e^{i2k\pi/5}$ is a geometric series with ratio $e^{i2\pi/5}$, which gives the sum of all fifth roots of unity except 1. Since the sum of all fifth roots of unity is 0, we have $\sum_{k=1}^{4}e^{i2k\pi/5} = -1$. Therefore, $-i \cdot (-1) = i$, and taking the principal value gives 1.
Correct Answer: 1