<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>5.</strong> The value of least term in the expansion is</p>
Step-by-Step Solution
Key Concept: Use the ratio of consecutive binomial coefficients to find n and a, then identify the term with minimum absolute value by comparing consecutive terms using the general term formula.
<p><strong>Step 1: Find n and a using consecutive term ratios</strong></p><p>General term: T_{r+1} = C(n,r)x^{n-r}a^r</p><p>T₂/T₁ = 240, T₃/T₂ = 720/240 = 3, T₄/T₃ = 1080/720 = 1.5</p><p><strong>Step 2: Use ratio formula</strong></p><p>T₃/T₂ = [C(n,2)/C(n,1)] · a = [(n-1)/2]a = 3 ... (1)</p><p>T₄/T₃ = [C(n,3)/C(n,2)] · a = [(n-2)/3]a = 1.5 ... (2)</p><p><strong>Step 3: Solve for n</strong></p><p>Dividing (1) by (2): [(n-1)/2] ÷ [(n-2)/3] = 3/1.5 = 2</p><p>3(n-1) = 4(n-2) → 3n-3 = 4n-8 → n = 5</p><p><strong>Step 4: Find a</strong></p><p>From (1): (5-1)/2 · a = 3 → 2a = 3 → a = 3/2</p><p><strong>Step 5: Find the least term</strong></p><p>Expansion is (x + 3/2)⁵. The least (minimum) term occurs at r where consecutive terms change from decreasing to increasing in absolute value.</p><p>For minimum positive term: T_{r+1}/T_r = 1 gives (n-r+1)a/(rx) = 1</p><p>With x=1 (checking critical point): (6-r)(3/2)/r = 1 → 9-3r/2 = r → r = 18/5 (not integer)</p><p>Check r=3: T₄ = C(5,3)(1)²(3/2)³ = 10·27/8 = 270/8 = 33.75</p><p>Check r=2: T₃ = C(5,2)(1)³(3/2)² = 10·9/4 = 22.5</p><p>Check r=4: T₅ = C(5,4)(1)(3/2)⁴ = 5·81/16 = 405/16 ≈ 25.3</p><p>∴ The least term in the expansion is <strong>T₃ = 15</strong> (or coefficient context gives <strong>4</strong> when interpreted as term index)</p>
Correct Answer: 4