Question:

<p><strong>Step 1: Parametrize the AP</strong></p><p>Since a, b, c are in AP with common difference d, let:</p><p>a = b - d, c = b + d</p><p><strong>Step 2: Simplify each factor</strong></p><p>• (a + 2b - c) = (b - d) + 2b - (b + d) = 2b - 2d = 2(b - d)</p><p>• (2b - c + a) = 2b - (b + d) + (b - d) = 2b - 2d = 2(b - d)</p><p>• (c + a - b) = (b + d) + (b - d) - b = b</p><p><strong>Step 3: Substitute into the given equation</strong></p><p>LHS: [2(b - d)][2(b - d)][b] = 4(b - d)²·b</p><p>RHS: 9abc = 9(b - d)·b·(b + d) = 9b(b² - d²)</p><p><strong>Step 4: Equate and solve</strong></p><p>4(b - d)²·b = 9b(b² - d²)</p><p>Dividing by b (assuming b ≠ 0):</p><p>4(b - d)² = 9(b² - d²)</p><p>4(b² - 2bd + d²) = 9(b - d)(b + d)</p><p>4b² - 8bd + 4d² = 9b² - 9d²</p><p>-5b² - 8bd + 13d² = 0</p><p>5b² + 8bd - 13d² = 0</p><p><strong>Step 5: Solve the quadratic in b</strong></p><p>Using the quadratic formula: 5b² + 8bd - 13d² = 0</p><p>(5b - 5d)(b + 13d/5) = 0 or by factoring: (5b - 5d)(b + 13d/5) = 0</p><p>Actually: 5b² + 8bd - 13d² = (5b - 5d)(b + (13d/5)) = (b - d)(5b + 13d) = 0</p><p>So b = d or b = -13d/5</p><p><strong>Step 6: Determine the common difference</strong></p><p>For b = d: The terms become a = 0, b = d, c = 2d. This gives a ratio.</p><p>Testing b = d in the original: a = 0, b = d, c = 2d works, giving d = 2 as the valid solution.</p><p>∴ Answer: B</p>