<p><strong>For Problems 4–6:</strong> The 2nd, 3rd, and 4th terms in the expansion of \((x + a)^n\) are 240, 720, and 1080, respectively.</p><p><strong>6.</strong> The sum of odd-numbered terms is</p>
Step-by-Step Solution
Key Concept: Once x, a, and n are determined from the given conditions, use the binomial expansion formula to identify odd-positioned terms (1st, 3rd, 5th, ...) and sum them using the identity that involves substituting specific values into the binomial expansion.
<p><strong>Step 1: Find x, a, and n from given conditions</strong></p><p>Given: T₂ = 240, T₃ = 720, T₄ = 1080</p><p>Using T_{r+1} = C(n,r)x^(n-r)a^r:</p><p><strong>Step 2: Use ratio method</strong></p><p>T₃/T₂ = [C(n,2)x^(n-2)a²]/[C(n,1)x^(n-1)a] = 720/240 = 3</p><p>⟹ (n-1)a/(2x) = 3 ... (i)</p><p>T₄/T₃ = [C(n,3)x^(n-3)a³]/[C(n,2)x^(n-2)a²] = 1080/720 = 1.5</p><p>⟹ (n-2)a/(3x) = 1.5 ... (ii)</p><p><strong>Step 3: Solve the system</strong></p><p>From (i) and (ii): n = 10, x = 2, a = 3</p><p><strong>Step 4: Find sum of odd-positioned terms</strong></p><p>For (2+3)¹⁰ = 5¹⁰ and (2-3)¹⁰ = (-1)¹⁰ = 1</p><p>Sum of odd-positioned terms = [(2+3)¹⁰ + (2-3)¹⁰]/2 = (5¹⁰ + 1)/2</p><p>Alternatively, S_odd = [5¹⁰ + 1]/2 = (9765625 + 1)/2 = 4882813</p><p>∴ Answer: (5¹⁰ + 1)/2 or 4882813</p>
Correct Answer: 1