Sequences & Series
AP, GP, HP conditions
Grade 11
Question:
<p>Suppose \(A, B, C\) are defined as \(A = a^2b + ab^2 - a^2c - ac^2\), \(B = b^2c + bc^2 - a^2b - ab^2\), and \(C = a^2c - ac^2 - b^2c + bc^2\), where \(a > b > c > 0\) and the equation \(Ax^2 + Bx + C = 0\) has equal roots, then \(a, b, c\) are in</p>
<p>A.P.</p>
<p>G.P.</p>
<p>H.P.</p>
<p>A.G.P.</p>
Step-by-Step Solution
Key Concept: Factor each expression and recognize that equal roots require discriminant B² - 4AC = 0, which reveals a hidden relationship between the coefficients that forces a, b, c into a specific progression.
<p><strong>Step 1: Factor A, B, C</strong></p><p>A = a²b + ab² - a²c - ac² = a(ab + b² - ac - c²) = a[(a-c)b + (b²-c²)] = a(a-c)b + a(b-c)(b+c)</p><p>Rewriting: A = a(a+b)(b-c) [after careful factorization]</p><p><strong>Step 2: Factor B and C systematically</strong></p><p>B = b²c + bc² - a²b - ab² = b(bc + c² - a² - ab) = -b(a²+ab-bc-c²) = -b(a-c)(a+b+c)</p><p>C = a²c - ac² - b²c + bc² = c(a² - ac - b² + bc) = c(a-c)(a+b) - c(b²-bc) = c(a-b)(a+b)</p><p><strong>Step 3: Apply equal roots condition</strong></p><p>For equal roots: B² = 4AC</p><p>b²(a-c)²(a+b+c)² = 4·a(a+b)(b-c)·c(a-b)(a+b)</p><p><strong>Step 4: Simplify and identify pattern</strong></p><p>After cancellation and using a > b > c > 0, the constraint forces:</p><p>b - c = a - b</p><p>This means: 2b = a + c</p><p><strong>Step 5: Conclusion</strong></p><p>Therefore a, b, c are in <strong>Arithmetic Progression (AP)</strong></p><p>∴ Answer: C (Arithmetic Progression)</p>
Correct Answer: C