If `z_(1),z_(2),z_(3),…………..,z_(n)` are n nth roots of unity, then for `k=1,2,,………,n`

Options

A. (a)`|z_(k)|=k|z_(k+1|`
B. (b)`|z_(k+1)|=k|z_(k1|`
C. (c)`|z_(k+1)|=|zk|+|z_(k-1)|`
D. (d)`|z_(k)|=|z_(k+1)|`

Answer

To solve the problem regarding the nth roots of unity, we will follow these steps:

### Step 1: Understanding nth Roots of Unity
The nth roots of unity are the complex numbers given by:
\[ z_k = e^{2\pi i k/n} \quad \text{for } k = 0, 1, 2, \ldots, n-1 \]
These roots lie on the unit circle in the complex plane, meaning that their modulus is 1.

### Step 2: Modulus of nth Roots of Unity
Since all the roots \( z_1, z_2, \ldots, z_n \) are on the unit circle, we can state:
\[ |z_k| = 1 \quad \text{for } k = 1, 2, \ldots, n \]
This means that the modulus of each root is equal to 1.

### Step 3: Analyzing the Given Condition
The problem states that for \( k = 1, 2, \ldots, n \):
\[ |z_k| = |z_{k+1}| \]
where \( z_{n+1} \) is understood to wrap around to \( z_1 \) (i.e., \( z_{n+1} = z_1 \)).

### Step 4: Conclusion
Since we have established that \( |z_k| = 1 \) for all \( k \), it follows that:
\[ |z_k| = |z_{k+1}| = 1 \]
Thus, the equality holds true for all \( k \).

### Final Answer
The correct conclusion is that for all \( k = 1, 2, \ldots, n \):
\[ |z_k| = 1 \]