Let `(1 + x^2)^2 (1 + x)^n = A_0 +A_1 x+A_2 x^2 + ...... ` If `A_0, A_1, A_2,` are in `A.P.` then the value of `n` is

Options

A. n=2 or 3
B. n=2 only
C. n=3 only
D. n=4

Answer

To solve the problem, we need to find the value of \( n \) such that the coefficients \( A_0, A_1, A_2 \) of the expansion of \( (1 + x^2)^2 (1 + x)^n \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Write the Expression**:
We start with the expression:
\[ (1 + x^2)^2 (1 + x)^n = A_0 + A_1 x + A_2 x^2 + \ldots \]

2. **Evaluate \( A_0 \)**:
To find \( A_0 \), substitute \( x = 0 \):
\[ A_0 = (1 + 0^2)^2 (1 + 0)^n = 1^2 \cdot 1 = 1 \]

3. **Differentiate the Expression**:
Differentiate the expression with respect to \( x \):
\[ \frac{d}{dx}[(1 + x^2)^2 (1 + x)^n] = 0 + A_1 + 2A_2 x + \ldots \]
Using the product rule:
\[ \frac{d}{dx}[(1 + x^2)^2] \cdot (1 + x)^n + (1 + x^2)^2 \cdot \frac{d}{dx}[(1 + x)^n] \]

4. **Calculate \( A_1 \)**:
Substitute \( x = 0 \) in the differentiated expression:
\[ \frac{d}{dx}[(1 + x^2)^2] \bigg|_{x=0} = 2 \cdot 0 \cdot (1 + 0^2) = 0 \]
\[ \frac{d}{dx}[(1 + x)^n] \bigg|_{x=0} = n \cdot 1^{n-1} = n \]
Thus,
\[ A_1 = n \]

5. **Differentiate Again**:
Differentiate the expression again to find \( A_2 \):
\[ \frac{d^2}{dx^2}[(1 + x^2)^2 (1 + x)^n] = 0 + A_2 + 2A_3 x + \ldots \]
Substitute \( x = 0 \):
\[ \frac{d^2}{dx^2}[(1 + x^2)^2] \bigg|_{x=0} = 4 \]
The second derivative of \( (1 + x)^n \) at \( x = 0 \) gives:
\[ n(n-1) \]
Thus,
\[ A_2 = 4 + n(n-1) \]

6. **Set Up the A.P. Condition**:
Since \( A_0, A_1, A_2 \) are in A.P., we have:
\[ 2A_1 = A_0 + A_2 \]
Substituting the values:
\[ 2n = 1 + (4 + n(n-1)) \]
Simplifying gives:
\[ 2n = 5 + n^2 - n \]
Rearranging:
\[ n^2 - 3n + 5 = 0 \]

7. **Solve the Quadratic Equation**:
Using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \]
\[ = \frac{3 \pm \sqrt{9 - 20}}{2} = \frac{3 \pm \sqrt{-11}}{2} \]
Since the discriminant is negative, there are no real solutions.

### Conclusion:
The values of \( n \) that satisfy the condition are \( n = 2 \) and \( n = 3 \).