<p>If the coefficients of the three successive terms in the binomial expansion of \((1+x)^n\) are in the ratio \(1:7:42\), then the first of these terms in the expansion is</p>
Step-by-Step Solution
Key Concept: If three successive binomial coefficients C(n,r), C(n,r+1), C(n,r+2) are in ratio 1:7:42, use the relationship between consecutive binomial coefficients: C(n,r+1)/C(n,r) = (n-r)/(r+1) to form equations and solve for n and r simultaneously.
<p><strong>Step 1:</strong> Let the three successive coefficients be C(n,r), C(n,r+1), C(n,r+2) in ratio 1:7:42.</p><p><strong>Step 2:</strong> From the ratio C(n,r+1)/C(n,r) = 7/1, we get: (n-r)/(r+1) = 7, so n - r = 7r + 7, giving <strong>n = 8r + 7</strong></p><p><strong>Step 3:</strong> From the ratio C(n,r+2)/C(n,r+1) = 42/7 = 6, we get: (n-r-1)/(r+2) = 6, so n - r - 1 = 6r + 12, giving <strong>n = 7r + 13</strong></p><p><strong>Step 4:</strong> Equating the two expressions: 8r + 7 = 7r + 13, so <strong>r = 6</strong></p><p><strong>Step 5:</strong> Substituting r = 6: n = 8(6) + 7 = <strong>55</strong></p><p><strong>Step 6:</strong> Verify: C(55,6) : C(55,7) : C(55,8) = 1 : 7 : 42 ✓</p><p><strong>Step 7:</strong> The first of these three terms is the (r+1)th term = <strong>6th term</strong> (or T₇ if counting from T₁)</p><p>∴ Answer: A</p>
Correct Answer: A