Let `(1+x^2)^2(1+x)^n=sum_(k=0)^(n+4)a_k x^k`. If `a_1`, `a_2` and `a_3` are in arithmetic progression, then the possible value/values of `n` is/are a. 5 b. 4 c. `3` d. `2`

Explanation

To solve the problem, we need to analyze the expression \((1+x^2)^2(1+x)^n\) and find the coefficients \(a_1\), \(a_2\), and \(a_3\) of \(x\), \(x^2\), and \(x^3\) respectively. We will then check the condition that these coefficients are in arithmetic progression.

### Step 1: Expand \((1+x^2)^2\)
First, we expand \((1+x^2)^2\):
\[ (1+x^2)^2 = 1 + 2x^2 + x^4 \]

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