<p>If <i>a</i>, <i>b</i>, and <i>c</i> are in A.P., then \(a^3 + c^3 - 8b^3\) is equal to</p>
Step-by-Step Solution
Key Concept: Since a, b, c are in A.P., we have b = (a+c)/2. Use this constraint to express a³ + c³ - 8b³ in terms of a relationship, then recognize the algebraic identity pattern for sum of cubes.
<p><strong>Step 1:</strong> Given a, b, c are in A.P., so 2b = a + c, or b = (a+c)/2</p><p><strong>Step 2:</strong> Calculate 8b³ = 8·((a+c)/2)³ = (a+c)³</p><p><strong>Step 3:</strong> Therefore: a³ + c³ - 8b³ = a³ + c³ - (a+c)³</p><p><strong>Step 4:</strong> Expand (a+c)³ = a³ + 3a²c + 3ac² + c³</p><p><strong>Step 5:</strong> Substitute: a³ + c³ - (a³ + 3a²c + 3ac² + c³) = -3a²c - 3ac² = -3ac(a+c)</p><p><strong>Step 6:</strong> Since a + c = 2b: a³ + c³ - 8b³ = -3ac(2b) = <strong>-6abc</strong></p><p>∴ Answer: D (which is -6abc)</p>
Correct Answer: D