<p>If the 4th term in the expansion of \((ax + 1/x)^n\) is 5/2, then</p>
Step-by-Step Solution
Key Concept: The general term in binomial expansion is T_(r+1) = C(n,r) × a^(n-r) × x^(n-r) × (1/x)^r. For the 4th term, r=3, and the power of x must be identified to separate coefficient from variable terms.
<p><strong>Step 1:</strong> Write the general term for (ax + 1/x)^n</p><p>T_(r+1) = C(n,r) × (ax)^(n-r) × (1/x)^r = C(n,r) × a^(n-r) × x^(n-r-r) = C(n,r) × a^(n-r) × x^(n-2r)</p><p><strong>Step 2:</strong> For the 4th term, r = 3</p><p>T₄ = C(n,3) × a^(n-3) × x^(n-6)</p><p><strong>Step 3:</strong> For T₄ to be a constant (independent of x), the power of x must be zero</p><p>n - 6 = 0 ⟹ n = 6</p><p><strong>Step 4:</strong> Substitute n = 6 into T₄</p><p>T₄ = C(6,3) × a^(6-3) = C(6,3) × a³ = 20a³</p><p><strong>Step 5:</strong> Set T₄ equal to 5/2</p><p>20a³ = 5/2</p><p>a³ = 5/40 = 1/8</p><p>a = 1/2</p><p><strong>Step 6:</strong> Verify: With n = 6 and a = 1/2, we get T₄ = 20 × (1/8) = 20/8 = 5/2 ✓</p><p>∴ Answer: n = 6, a = 1/2 (AB)</p>
Correct Answer: AB