<p>If \(1+x^4+x^5=\displaystyle\sum_{i=0}^{5}a_i(1+x)^i\) for all \(x\in R\), then \(a_2\) is</p>
Step-by-Step Solution
Key Concept: Expand the right side using binomial theorem and match coefficients of like powers of x on both sides to find a₂. The left side 1+x⁴+x⁵ must equal the linear combination of binomial expansions on the right.
<p><strong>Step 1:</strong> Expand the right side using binomial theorem:</p><p>∑ᵢ₌₀⁵ aᵢ(1+x)ⁱ = a₀ + a₁(1+x) + a₂(1+x)² + a₃(1+x)³ + a₄(1+x)⁴ + a₅(1+x)⁵</p><p><strong>Step 2:</strong> Expand each binomial term and collect coefficients of powers of x:</p><p>Coefficient of x⁰: a₀ + a₁ + a₂ + a₃ + a₄ + a₅ = 1</p><p>Coefficient of x¹: a₁ + 2a₂ + 3a₃ + 4a₄ + 5a₅ = 0</p><p>Coefficient of x²: a₂ + 3a₃ + 6a₄ + 10a₅ = 0</p><p>Coefficient of x³: a₃ + 4a₄ + 10a₅ = 0</p><p>Coefficient of x⁴: a₄ + 5a₅ = 1</p><p>Coefficient of x⁵: a₅ = 1</p><p><strong>Step 3:</strong> Solve backwards:</p><p>From x⁵: a₅ = 1</p><p>From x⁴: a₄ + 5(1) = 1 ⟹ a₄ = -4</p><p>From x³: a₃ + 4(-4) + 10(1) = 0 ⟹ a₃ = 6</p><p>From x²: a₂ + 3(6) + 6(-4) + 10(1) = 0 ⟹ a₂ + 18 - 24 + 10 = 0 ⟹ a₂ = -4</p><p><strong>∴ Answer: a₂ = -4</strong></p>
Correct Answer: C