<p>Find the middle term in the expansion of <span>\(\left(x^2 + \dfrac{1}{x^2} + 2\right)^n\)</span>.</p>
Step-by-Step Solution
Key Concept: Recognize that (x² + 1/x² + 2)ⁿ = (x + 1/x)²ⁿ by rewriting the expression as a perfect square, then apply the binomial theorem to find the middle term of the expanded form.
<p><strong>Step 1:</strong> Rewrite the expression by recognizing the perfect square pattern:</p><p>x² + 1/x² + 2 = x² + 1/x² + 2·x·(1/x) = (x + 1/x)²</p><p><strong>Step 2:</strong> Substitute into the original expression:</p><p>(x² + 1/x² + 2)ⁿ = [(x + 1/x)²]ⁿ = (x + 1/x)²ⁿ</p><p><strong>Step 3:</strong> Apply binomial theorem to (x + 1/x)²ⁿ:</p><p>General term: Tₖ₊₁ = C(2n,k)·xᵏ·(1/x)²ⁿ⁻ᵏ = C(2n,k)·x^(2k-2n)</p><p><strong>Step 4:</strong> For the middle term in expansion of (x + 1/x)²ⁿ (which has 2n+1 terms), the middle term is the (n+1)th term:</p><p>T_{n+1} = C(2n,n)·x⁰ = C(2n,n) = (2n)!/(n!)²</p><p><strong>Step 5:</strong> The coefficient of the middle term is:</p><p>∴ Answer: <strong>(2n)!/(n!)²</strong></p>
Correct Answer: (2n)!/(n!)²