The value of the expression `1.(2-omega).(2-omega^2)+2.(3-omega)(3-omega^2)+.+(n-1)(n-omega)(n-omega^2),` where omega is an imaginary cube root of unity, is………

Explanation

To solve the expression \( S = 1 \cdot (2 - \omega)(2 - \omega^2) + 2 \cdot (3 - \omega)(3 - \omega^2) + \ldots + (n-1) \cdot (n - \omega)(n - \omega^2) \), where \( \omega \) is an imaginary cube root of unity, we can follow these steps:

### Step 1: Understand the properties of \( \omega \)
The cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3} \]
These satisfy the equations:
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