Suppose `omega ne 1` is cube root of unity. If `1(2-omega) (2-omega^(2)) + 2(3 - omega) (3 - omega^(2)) + ….+ (n-1)(n-omega) (n-omega^(2)) = 33`, then n is equal to ____________

Answer

To solve the problem, we need to evaluate the expression given and find the value of \( n \) such that:

\[ 1(2 - \omega)(2 - \omega^2) + 2(3 - \omega)(3 - \omega^2) + \ldots + (n - 1)(n - \omega)(n - \omega^2) = 33 \]

where \( \omega \) is a cube root of unity and \( \omega \neq 1 \).

### Step 1: Understanding the Cube Roots of Unity
The cube roots of unity are \( 1, \omega, \omega^2 \) where:
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3} \]
They satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
Also, we have:
\[ \omega^3 = 1 \]

### Step 2: Simplifying the Expression
The expression can be rewritten as:
\[ \sum_{r=1}^{n-1} r (r + 1 - \omega)(r + 1 - \omega^2) \]
This can be expanded as:
\[ \sum_{r=1}^{n-1} r \left( (r + 1)^2 - (r + 1)(\omega + \omega^2) + \omega \omega^2 \right) \]
Since \( \omega + \omega^2 = -1 \) and \( \omega \omega^2 = \omega^3 = 1 \), we have:
\[ (r + 1)^2 + r(r + 1) \]

### Step 3: Evaluating the Summation
The expression becomes:
\[ \sum_{r=1}^{n-1} r \left( (r + 1)^2 + r(r + 1) \right) \]
This simplifies to:
\[ \sum_{r=1}^{n-1} r \left( r^2 + 2r + 1 + r^2 + r \right) = \sum_{r=1}^{n-1} r (2r^2 + 3r + 1) \]

### Step 4: Finding the Value of \( n \)
We need to evaluate:
\[ \sum_{r=1}^{n-1} r (2r^2 + 3r + 1) = 33 \]
Calculating the left-hand side:
1. The sum of \( r \) from \( 1 \) to \( n-1 \) is \( \frac{(n-1)n}{2} \).
2. The sum of \( r^2 \) from \( 1 \) to \( n-1 \) is \( \frac{(n-1)n(2(n-1)+1)}{6} \).

Using these formulas, we can calculate the left-hand side and set it equal to 33.

### Step 5: Trial and Error for \( n \)
We can try different integer values for \( n \):
- For \( n = 3 \):
\[ 1(2 - \omega)(2 - \omega^2) + 2(3 - \omega)(3 - \omega^2) = 1(2 - \omega)(2 - \omega^2) + 2(3 - \omega)(3 - \omega^2) \]
Calculating:
\[ = 1 \cdot (2 - \omega)(2 - \omega^2) + 2 \cdot (3 - \omega)(3 - \omega^2) \]
This can be computed to check if it equals 33.

After testing values, we find that \( n = 3 \) satisfies the equation.

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{3} \]