The value of `alpha^(-n)+alpha^(-2n), n in N` and `alpha` is a non-real cube root of unity, is

Options

A. 3, if n is a multiple of 3
B. `-1`, if n is a mulitiple of 3
C. 2, if n is a multiple of 3
D. none of these

Explanation

To solve the problem of finding the value of \( \alpha^{-n} + \alpha^{-2n} \) where \( \alpha \) is a non-real cube root of unity and \( n \in \mathbb{N} \), we can follow these steps:

### Step 1: Define the Non-Real Cube Roots of Unity
The non-real cube roots of unity are given by:
\[ \alpha = \omega = e^{2\pi i / 3} \quad \text{and} \quad \alpha^2 = \omega^2 = e^{-2\pi i / 3} \]
where \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

### Step 2: Rewrite the Expression
We need to find:
\[ \alpha^{-n} + \alpha^{-2n} \]
Since \( \alpha = \omega \), we can rewrite this as:
\[ \omega^{-n} + \omega^{-2n} = \frac{1}{\omega^n} + \frac{1}{\omega^{2n}} \]

### Step 3: Simplify the Expression
Using the property of cube roots of unity, we know:
\[ \frac{1}{\omega^n} = \omega^{3-k} \quad \text{if } n = 3k + r \text{ where } r = 0, 1, 2 \]
Thus, we can express:
\[ \frac{1}{\omega^n} = \omega^{3k} \cdot \omega^{-r} = \omega^{-r} \quad \text{(since } \omega^3 = 1\text{)} \]
Similarly,
\[ \frac{1}{\omega^{2n}} = \omega^{-2r} \]

### Step 4: Evaluate Based on the Value of \( n \)
Now, we evaluate \( \omega^{-r} + \omega^{-2r} \) based on the value of \( r \):
1. **If \( n = 3k \)** (i.e., \( r = 0 \)):
\[ \omega^{-0} + \omega^{-0} = 1 + 1 = 2 \]
2. **If \( n = 3k + 1 \)** (i.e., \( r = 1 \)):
\[ \omega^{-1} + \omega^{-2} = \omega^2 + \omega = -1 \quad \text{(since } 1 + \omega + \omega^2 = 0\text{)} \]
3. **If \( n = 3k + 2 \)** (i.e., \( r = 2 \)):
\[ \omega^{-2} + \omega^{-1} = \omega + \omega^2 = -1 \]

### Final Result
Thus, we conclude:
- If \( n \) is a multiple of 3, \( \alpha^{-n} + \alpha^{-2n} = 2 \).
- If \( n \) is not a multiple of 3, \( \alpha^{-n} + \alpha^{-2n} = -1 \).

### Summary
The value of \( \alpha^{-n} + \alpha^{-2n} \) is:
\[ \begin{cases} 2 & \text{if } n \text{ is a multiple of } 3 \\ -1 & \text{if } n \text{ is not a multiple of } 3 \end{cases} \]