Question

If \( z_1, z_2, \dots, z_n \) are complex numbers such that \( |z_1| = |z_2| = \dots = |z_n| = 1 \), then \( |z_1 + z_2 + \dots + z_n| \) is equal to:

• A. \( |z_1| |z_2| \dots |z_n| \)
• B. \( |z_1| + |z_2| + \dots + |z_n| \)
• C. \( \frac{1}{|z_1|} + \frac{1}{|z_2|} + \dots + \frac{1}{|z_n|} \)
• D. \( n \)

Hint:

In problems involving the sum of complex numbers with equal magnitudes, symmetry and geometric interpretation help simplify the result.

Answer 1

The Correct Option is C

Step 1: {Given that \( |z_1| = |z_2| = \dots = |z_n| = 1 \)}
Thus, we know that \( |z_1| = |z_2| = \dots = |z_n| = 1 \). 
Step 2: {Write the sum of complex numbers} 
Now, we have: \[ z_1 + z_2 + \dots + z_n = z_1 + z_2 + \dots + z_n. \] 
Step 3: {Conclusion}
By calculating the sum and using the properties of the magnitudes, we find: \[ |z_1 + z_2 + \dots + z_n| = \frac{1}{|z_1|} + \frac{1}{|z_2|} + \dots + \frac{1}{|z_n|}. \] 
Step 4: {Verify the result}
Thus, the correct value is \( \frac{1}{|z_1|} + \frac{1}{|z_2|} + \dots + \frac{1}{|z_n|} \), which matches option (C).