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

Options

A. `(n(n^(2)+2))/(3)`
B. `(n(n^(2)-2))/(3)`
C. `(n(n^(2)+1))/(3)`
D. `(n(n^(2)-1))/(3)`

Answer

To solve the expression

\[ S = (1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}) + (2 + \frac{1}{\omega})(2 + \frac{1}{\omega^2}) + (3 + \frac{1}{\omega})(3 + \frac{1}{\omega^2}) + \ldots + (n + \frac{1}{\omega})(n + \frac{1}{\omega^2}), \]

where \(\omega\) is an imaginary cube root of unity, we will proceed step by step.

### Step 1: Understanding \(\omega\)
The cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3}. \]
They satisfy the equations:
\[ 1 + \omega + \omega^2 = 0 \quad \text{and} \quad \omega^3 = 1. \]

### Step 2: Simplifying the Terms
We can rewrite each term in the summation:
\[ (1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}) = (1 + \frac{1}{\omega})(1 + \frac{1}{\omega^2}). \]
Using the property of \(\omega\):
\[ \frac{1}{\omega} = \omega^2 \quad \text{and} \quad \frac{1}{\omega^2} = \omega, \]
we can rewrite:
\[ (1 + \omega^2)(1 + \omega). \]

### Step 3: Expanding the Terms
Now, we expand:
\[ (1 + \omega)(1 + \omega^2) = 1 + \omega + \omega^2 + \omega \cdot \omega^2 = 1 + (-1) + \omega \cdot \omega^2 = 1 - 1 + \omega^3 = 1. \]
Thus, each term simplifies to 1.

### Step 4: Summing the Terms
Now, we can express the entire sum \(S\):
\[ S = \sum_{r=1}^{n} (r + \frac{1}{\omega})(r + \frac{1}{\omega^2}) = \sum_{r=1}^{n} 1 = n. \]

### Step 5: Final Expression
Therefore, the value of the expression is:
\[ S = n. \]

### Conclusion
The final answer is:
\[ \boxed{n}. \]