<p>If some three consecutive coefficients in the binomial expansion of \((x + 1)^n\) in powers of \(x\) are in the ratio \(2 : 15 : 70\), then the average of these three coefficients is ___________.</p>
Step-by-Step Solution
Key Concept: Three consecutive binomial coefficients C(n,r), C(n,r+1), C(n,r+2) satisfy the ratio condition. Use the relationship between consecutive binomial coefficients: C(n,r+1)/C(n,r) = (n-r)/(r+1) to set up equations from the given ratio.
<p><strong>Step 1:</strong> Let the three consecutive coefficients be C(n,r), C(n,r+1), C(n,r+2) in ratio 2:15:70.</p><p>Set C(n,r) = 2k, C(n,r+1) = 15k, C(n,r+2) = 70k for some constant k.</p><p><strong>Step 2:</strong> Use the ratio formula: C(n,r+1)/C(n,r) = (n-r)/(r+1) = 15k/2k = 15/2</p><p>This gives: 2(n-r) = 15(r+1) → 2n - 2r = 15r + 15 → 2n = 17r + 15 ... (1)</p><p><strong>Step 3:</strong> Similarly, C(n,r+2)/C(n,r+1) = (n-r-1)/(r+2) = 70k/15k = 14/3</p><p>This gives: 3(n-r-1) = 14(r+2) → 3n - 3r - 3 = 14r + 28 → 3n = 17r + 31 ... (2)</p><p><strong>Step 4:</strong> From equation (1): 2n = 17r + 15, so n = (17r + 15)/2</p><p>Substitute into equation (2): 3·(17r + 15)/2 = 17r + 31</p><p>→ 51r + 45 = 34r + 62 → 17r = 17 → r = 1</p><p><strong>Step 5:</strong> From equation (1): 2n = 17(1) + 15 = 32 → n = 16</p><p><strong>Step 6:</strong> The three coefficients are C(16,1) = 16, C(16,2) = 120, C(16,3) = 560</p><p>Verify ratio: 16:120:560 = 2:15:70 ✓</p><p><strong>Step 7:</strong> Average of three coefficients = (16 + 120 + 560)/3 = 696/3 = 232</p><p>∴ Answer: 232</p>
Correct Answer: 964