<p>Let \((1+x^2)^2(1+x)^n = \displaystyle\sum_{k=0}^{n+4} a_k x^k\). If \(a_1,\ a_2\) and \(a_3\) are in arithmetic progression, then the possible value/values of \(n\) is/are</p>
Step-by-Step Solution
Key Concept: Expand (1+x²)²(1+x)ⁿ to find coefficients a₁, a₂, a₃ explicitly, then apply the AP condition 2a₂ = a₁ + a₃ to form a quadratic equation in n.
<p><strong>Step 1:</strong> Expand (1+x²)² = 1 + 2x² + x⁴</p><p><strong>Step 2:</strong> The product is (1 + 2x² + x⁴)(1+x)ⁿ where (1+x)ⁿ = Σ C(n,r)xʳ</p><p><strong>Step 3:</strong> Identify coefficients:</p><ul><li>a₁ = C(n,1) = n</li><li>a₂ = C(n,2) + 2·C(n,0) = n(n-1)/2 + 2</li><li>a₃ = C(n,3) + 2·C(n,1) = n(n-1)(n-2)/6 + 2n</li></ul><p><strong>Step 4:</strong> Apply AP condition: 2a₂ = a₁ + a₃</p><p>2[n(n-1)/2 + 2] = n + [n(n-1)(n-2)/6 + 2n]</p><p>n(n-1) + 4 = 3n + n(n-1)(n-2)/6</p><p>Multiply by 6: 6n(n-1) + 24 = 18n + n(n-1)(n-2)</p><p>6n² - 6n + 24 = 18n + n³ - 3n² + 2n</p><p>0 = n³ - 9n² + 26n - 24</p><p><strong>Step 5:</strong> Factor: (n-2)(n-3)(n-4) = 0</p><p>Solutions: n = 2, 3, 4</p><p><strong>Step 6:</strong> Verify all are valid (n ≥ 0, n ∈ ℤ). All three satisfy domain constraints.</p><p>∴ Answer: <strong>n = 2, 3, 4</strong> (Options A, C, D)</p>
Correct Answer: ACD