<p>The coefficients of three consecutive terms of \((1+x)^{n+5}\) are in the ratio \(5:10:14\). Then \(n =\) _______.</p>
Step-by-Step Solution
Key Concept: Use the ratio property of binomial coefficients: C(n,r)/C(n,r+1) = (r+1)/(n-r) to set up equations from consecutive term coefficients. The key is recognizing that if three consecutive binomial coefficients are in ratio 5:10:14, you can write two independent equations.
<p><strong>Step 1:</strong> Let the three consecutive binomial coefficients be C(n+5,r), C(n+5,r+1), C(n+5,r+2) in ratio 5:10:14.</p><p><strong>Step 2:</strong> Using the ratio formula: C(n+5,r)/C(n+5,r+1) = (r+1)/(n+5-r) = 5/10 = 1/2</p><p>This gives: 2(r+1) = n+5-r, so 3r+2 = n+5, hence n = 3r-3 ... (i)</p><p><strong>Step 3:</strong> Using C(n+5,r+1)/C(n+5,r+2) = (r+2)/(n+4-r) = 10/14 = 5/7</p><p>This gives: 7(r+2) = 5(n+4-r), so 12r+14 = 5n+20, hence 5n = 12r-6 ... (ii)</p><p><strong>Step 4:</strong> Substitute n = 3r-3 into equation (ii):</p><p>5(3r-3) = 12r-6</p><p>15r-15 = 12r-6</p><p>3r = 9</p><p>r = 3</p><p><strong>Step 5:</strong> Therefore n = 3(3)-3 = 9-3 = 6</p><p><strong>Verification:</strong> C(11,3):C(11,4):C(11,5) = 165:330:462 = 5:10:14 ✓</p><p>∴ Answer: <strong>6</strong></p>
Correct Answer: 6