<p>If the coefficients of <span>\((r-5)\)</span>th and <span>\((2r-1)\)</span>th terms in the expansion of <span>\((1+x)^{34}\)</span> are equal, find <span>\(r\)</span>.</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (1+x)^n, the coefficient of the kth term is C(n,k-1). Two terms have equal coefficients when their position indices satisfy either p+q=n or p=q.
<p><strong>Step 1:</strong> Identify the coefficient formula. The rth term in expansion of (1+x)^34 is C(34,r-1)·x^(r-1), so its coefficient is C(34,r-1).</p><p><strong>Step 2:</strong> Set up the equation. Coefficient of (r-5)th term = C(34,r-6). Coefficient of (2r-1)th term = C(34,2r-2).</p><p><strong>Step 3:</strong> Apply the condition C(34,r-6) = C(34,2r-2). This holds when either:<br/>(a) r-6 = 2r-2, or<br/>(b) (r-6) + (2r-2) = 34</p><p><strong>Step 4 (Case a):</strong> r-6 = 2r-2 ⟹ r = -4 (invalid, as r must be positive)</p><p><strong>Step 5 (Case b):</strong> (r-6) + (2r-2) = 34 ⟹ 3r - 8 = 34 ⟹ 3r = 42 ⟹ r = 14</p><p><strong>Step 6:</strong> Verify: (r-5)th term is 9th term with coefficient C(34,8). (2r-1)th term is 27th term with coefficient C(34,26). Since 8 + 26 = 34, we have C(34,8) = C(34,26) ✓</p><p>∴ Answer: r = 14</p>
Correct Answer: r = 14