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

### Step 1: Understand the properties of \(\omega\)
The imaginary cube roots of unity are the solutions to the equation \(x^3 - 1 = 0\). The roots are \(1\), \(\omega\), and \(\omega^2\), where:
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3} \]
We also know that:
...