<p>Divide 20 in four parts which are in AP such that the product of the first and the fourth is to the product of the second and the third = 2 : 3.</p>
Step-by-Step Solution
Key Concept: Use the symmetric property of AP where if four terms are in AP, express them as (a-3d), (a-d), (a+d), (a+3d) so their sum gives 4a=20, then use the product ratio condition to find d.
<p><strong>Step 1:</strong> Let the four parts in AP be (a-3d), (a-d), (a+d), (a+3d).</p><p><strong>Step 2:</strong> Sum condition: (a-3d) + (a-d) + (a+d) + (a+3d) = 20 → 4a = 20 → a = 5</p><p><strong>Step 3:</strong> Product ratio condition: [(a-3d)(a+3d)]/[(a-d)(a+d)] = 2/3</p><p><strong>Step 4:</strong> Using difference of squares: (a²-9d²)/(a²-d²) = 2/3</p><p><strong>Step 5:</strong> Substitute a = 5: (25-9d²)/(25-d²) = 2/3</p><p><strong>Step 6:</strong> Cross-multiply: 3(25-9d²) = 2(25-d²) → 75-27d² = 50-2d² → 25d² = 25 → d = ±1</p><p><strong>Step 7:</strong> For d = 1: parts are 2, 4, 6, 8</p><p>For d = -1: parts are 8, 6, 4, 2</p><p><strong>Verification:</strong> 2+4+6+8 = 20 ✓ and (2×8):(4×6) = 16:24 = 2:3 ✓</p><p>∴ <strong>Answer: 2, 4, 6, 8</strong></p>
Correct Answer: 2