<p>If the middle term in the expansion of \((x^2 + 1/x)^n\) is \(924x^6\), then find the value of \(n\).</p>
Step-by-Step Solution
Key Concept: For even n, the middle term is the (n/2 + 1)th term. Use the general term formula and match the power of x with the coefficient to find n.
<p><strong>Step 1:</strong> For the expansion of (x² + 1/x)^n, the general term is:</p><p>T_r = C(n, r-1) × (x²)^(n-r+1) × (1/x)^(r-1) = C(n, r-1) × x^(2n-2r+2) × x^(-(r-1))</p><p>T_r = C(n, r-1) × x^(2n-3r+3)</p><p><strong>Step 2:</strong> For even n, the middle term is T_(n/2 + 1), where r = n/2 + 1:</p><p>Power of x: 2n - 3(n/2 + 1) + 3 = 2n - 3n/2 - 3 + 3 = n/2</p><p>Given that power of x is 6: n/2 = 6 ⟹ n = 12</p><p><strong>Step 3:</strong> Verify the coefficient:</p><p>Middle term is T_7 = C(12, 6) × x^6</p><p>C(12, 6) = 924 ✓</p><p>∴ Answer: n = 12</p>
Correct Answer: 12