<p>Find the two successive terms in the expansion of <span>\((1 + x)^{24}\)</span> whose coefficients are in the ratio <span>\(1 : 4\)</span>.</p>
Step-by-Step Solution
Key Concept: For binomial expansion of (1+x)^n, the (r+1)th term has coefficient C(n,r). Set up the ratio of consecutive binomial coefficients C(24,r):C(24,r+1) = 1:4 and solve for r using the relationship C(n,r+1)/C(n,r) = (n-r)/(r+1).
<p><strong>Step 1:</strong> Let the two successive terms be the (r+1)th and (r+2)th terms in the expansion of (1+x)^24. Their coefficients are C(24,r) and C(24,r+1) respectively.</p><p><strong>Step 2:</strong> Given that the coefficients are in ratio 1:4, we have:</p><p>C(24,r) : C(24,r+1) = 1 : 4</p><p><strong>Step 3:</strong> This gives us: C(24,r+1)/C(24,r) = 4</p><p><strong>Step 4:</strong> Using the formula C(n,r+1)/C(n,r) = (n-r)/(r+1):</p><p>(24-r)/(r+1) = 4</p><p><strong>Step 5:</strong> Solving: 24 - r = 4(r+1)</p><p>24 - r = 4r + 4</p><p>20 = 5r</p><p>r = 4</p><p><strong>Step 6:</strong> The two successive terms are the (r+1)th = 5th term and (r+2)th = 6th terms.</p><p>∴ Answer: <strong>5th and 6th terms</strong></p>
Correct Answer: 5th and 6th terms