<p>The first three terms in the expansion of <span>\((1 + ax)^n\)</span> <span>\((n \neq 0)\)</span> are <span>\(1, 6x,\)</span> and <span>\(16x^2\)</span>. Then find the value of <span>\(a\)</span> and <span>\(n\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the ratio of consecutive binomial coefficients to relate them: the ratio of the (r+1)th term to the rth term equals [(n-r+1)·ax]/r. Setting up ratios for consecutive terms eliminates n, allowing you to solve for a directly.
<p><strong>Step 1:</strong> Write the first three terms using binomial expansion:</p><p>T₁ = 1</p><p>T₂ = C(n,1)·ax = n·ax = 6x</p><p>T₃ = C(n,2)·(ax)² = [n(n-1)/2]·a²x² = 16x²</p><p></p><p><strong>Step 2:</strong> From T₁ and T₂:</p><p>n·a = 6 ... (i)</p><p></p><p><strong>Step 3:</strong> From T₂ and T₃, use the ratio method:</p><p>T₃/T₂ = [n(n-1)/2]·a²x²/(n·ax) = [(n-1)·a]/2 = 16x²/(6x) = (8/3)x</p><p>This gives: [(n-1)·a]/2 = 8/3</p><p>(n-1)·a = 16/3 ... (ii)</p><p></p><p><strong>Step 4:</strong> Subtract equation (ii) from equation (i):</p><p>n·a - (n-1)·a = 6 - 16/3</p><p>a = 18/3 - 16/3 = 2/3</p><p></p><p><strong>Step 5:</strong> Substitute a = 2/3 into equation (i):</p><p>n·(2/3) = 6</p><p>n = 9</p><p></p><p><strong>Verification:</strong> T₂ = 9·(2/3)·x = 6x ✓ and T₃ = [9·8/2]·(4/9)·x² = 36·(4/9)·x² = 16x² ✓</p><p></p><p>∴ <strong>a = 2/3, n = 9</strong></p>
Correct Answer: a = 2/3, n = 9