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). 
 

Answer 2

Step 1: Given condition
All complex numbers satisfy \(|z_1| = |z_2| = \dots = |z_n| = 1\).

Step 2: Expression to find
We need to find the value of \(|z_1 + z_2 + \dots + z_n|\).

Step 3: Using the magnitude condition
Since each \(|z_k| = 1\), their magnitudes are equal to 1.
Therefore, \(\frac{1}{|z_k|} = 1\) for each \(k\).

Step 4: Sum of reciprocals of magnitudes
\[ \frac{1}{|z_1|} + \frac{1}{|z_2|} + \dots + \frac{1}{|z_n|} = 1 + 1 + \dots + 1 = n \]

Step 5: Result interpretation
Since all \(|z_k| = 1\), the sum \(|z_1 + z_2 + \dots + z_n|\) can be as large as \(n\) if all \(z_k\) point in the same direction.
So, the maximum possible value is:
\[ |z_1 + z_2 + \dots + z_n| = n = \frac{1}{|z_1|} + \frac{1}{|z_2|} + \dots + \frac{1}{|z_n|} \]

Final Answer: \(\frac{1}{|z_1|} + \frac{1}{|z_2|} + \dots + \frac{1}{|z_n|}\)