If `(1 + x + x^2)^n = (C_0 + C_1x + C_2 x^2 + ............)` then the value of `C_0 C_1-C_1C_2+C_2C_3.....`

Options

A. `3^(n)`
B. `(-1)^(n)`
C. `2^(n)`
D. none of these

Answer

To solve the problem, we need to find the value of the expression \( C_0 C_1 - C_1 C_2 + C_2 C_3 - \ldots \), where \( (1 + x + x^2)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + \ldots \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
We have \( (1 + x + x^2)^n \) expanded into a power series, where \( C_k \) are the coefficients of \( x^k \).

2. **Define the Expression \( S \)**:
Let \( S = C_0 C_1 - C_1 C_2 + C_2 C_3 - C_3 C_4 + \ldots \).

3. **Substituting \( x \) with \( -\frac{1}{x} \)**:
Replace \( x \) in the original equation with \( -\frac{1}{x} \):
\[ (1 - \frac{1}{x} + \frac{1}{x^2})^n = C_0 - C_1 \frac{1}{x} + C_2 \frac{1}{x^2} - C_3 \frac{1}{x^3} + \ldots \]

4. **Simplifying the Left-Hand Side**:
The left-hand side becomes:
\[ \left(\frac{x^2 - x + 1}{x^2}\right)^n = \frac{(x^2 - x + 1)^n}{x^{2n}} \]

5. **Multiplying by \(-1\)**:
Multiply both sides by \(-1\):
\[ -\frac{(x^2 - x + 1)^n}{x^{2n}} = -C_0 + C_1 \frac{1}{x} - C_2 \frac{1}{x^2} + C_3 \frac{1}{x^3} - \ldots \]

6. **Identifying Coefficients**:
From the expression, we can see that \( S \) corresponds to the coefficient of \( x^{-1} \) in the expansion of the left-hand side.

7. **Multiplying the Two Equations**:
Now, multiply the original equation and the modified equation:
\[ (1 + x + x^2)^n \cdot \left(-\frac{(x^2 - x + 1)^n}{x^{2n}}\right) \]
This will help us find the coefficient of \( x^{-1} \).

8. **Finding the Coefficient of \( x^{-1} \)**:
After simplification, we find that there are no terms of \( x^{-1} \) in the resulting expression. Therefore, the coefficient of \( x^{-1} \) is \( 0 \).

9. **Conclusion**:
Thus, the value of \( S \) is:
\[ S = 0 \]

### Final Answer:
The value of \( C_0 C_1 - C_1 C_2 + C_2 C_3 - \ldots \) is \( 0 \).