<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>4.</strong> The value of \((x - a)^n\) can be</p>
Step-by-Step Solution
Key Concept: Use the general term of binomial expansion to express the 2nd, 3rd, and 4th terms, then set up equations using the given values to find x, a, and n. Once these parameters are determined, evaluate (x - a)^n.
<p><strong>Step 1: Write the general terms</strong></p><p>In the expansion of (x + a)^n, the (r+1)th term is:</p><p>T_{r+1} = C(n,r) × x^{n-r} × a^r</p><p>• 2nd term (r=1): T_2 = C(n,1) × x^{n-1} × a = n·x^{n-1}·a = 240</p><p>• 3rd term (r=2): T_3 = C(n,2) × x^{n-2} × a^2 = [n(n-1)/2]·x^{n-2}·a^2 = 720</p><p>• 4th term (r=3): T_4 = C(n,3) × x^{n-3} × a^3 = [n(n-1)(n-2)/6]·x^{n-3}·a^3 = 1080</p><p><strong>Step 2: Find ratios of consecutive terms</strong></p><p>T_3/T_2 = {[n(n-1)/2]·x^{n-2}·a^2} / {n·x^{n-1}·a} = [(n-1)/2] · (a/x) = 720/240 = 3</p><p>Therefore: (n-1)·a/(2x) = 3 → (n-1)·a = 6x ... (Equation 1)</p><p>T_4/T_3 = {[n(n-1)(n-2)/6]·x^{n-3}·a^3} / {[n(n-1)/2]·x^{n-2}·a^2} = [(n-2)/3] · (a/x) = 1080/720 = 3/2</p><p>Therefore: (n-2)·a/(3x) = 3/2 → (n-2)·a = 9x/2 ... (Equation 2)</p><p><strong>Step 3: Solve for n</strong></p><p>From Equation 1: (n-1)·a = 6x</p><p>From Equation 2: (n-2)·a = 9x/2</p><p>Dividing: [(n-1)/(n-2)] = [6x/(9x/2)] = [6x × 2/(9x)] = 12/9 = 4/3</p><p>3(n-1) = 4(n-2)</p><p>3n - 3 = 4n - 8</p><p>n = 5</p><p><strong>Step 4: Find a and x</strong></p><p>From Equation 1: (5-1)·a = 6x → 4a = 6x → a = 3x/2</p><p>Substituting into T_2 = 240:</p><p>n·x^{n-1}·a = 5·x^4·(3x/2) = 240</p><p>(15/2)·x^5 = 240</p><p>x^5 = 32 → x = 2</p><p>Therefore: a = 3(2)/2 = 3</p><p><strong>Step 5: Calculate (x - a)^n</strong></p><p>(x - a)^n = (2 - 3)^5 = (-1)^5 = -1</p><p>∴ Answer: 3</p>
Correct Answer: 3