<p>If \((4x^2 + 1)^n = \displaystyle\sum_{r=0}^{n} a_r(1+x^2)^{n-r} x^{2r}\), then the value of \(\displaystyle\sum_{r=0}^{n} a_r\) is</p>
Step-by-Step Solution
Key Concept: To find Σaᵣ, substitute x = 1 into both sides of the given expansion equation. This converts the binomial expression into a numerical equation where coefficients become additive.
<p><strong>Step 1:</strong> Substitute x = 1 into the given equation:</p><p>(4(1)² + 1)ⁿ = Σ aᵣ(1+1)ⁿ⁻ʳ · 1²ʳ</p><p><strong>Step 2:</strong> Simplify the left side:</p><p>(4 + 1)ⁿ = 5ⁿ</p><p><strong>Step 3:</strong> Simplify the right side:</p><p>5ⁿ = Σ aᵣ · 2ⁿ⁻ʳ</p><p><strong>Step 4:</strong> To isolate Σaᵣ, substitute x = 0 into the original equation:</p><p>(0 + 1)ⁿ = Σ aᵣ(1)ⁿ⁻ʳ · 0</p><p>This gives: 1 = a₀(1)ⁿ, so a₀ = 1</p><p><strong>Step 5:</strong> Actually, differentiate or use the constraint directly. Setting x = 0 in the original: 1ⁿ = a₀ · 1ⁿ, giving a₀ = 1. Comparing coefficients of the expansion (4x² + 1)ⁿ with the RHS form and summing all terms by setting x = 1:</p><p>5ⁿ = Σ aᵣ · 2ⁿ⁻ʳ implies Σaᵣ = 5ⁿ/2ⁿ = (5/2)ⁿ</p><p>∴ Answer: <strong>(5/2)ⁿ</strong></p>
Correct Answer: A