<p>For an increasing A.P. \(a_1, a_2, \ldots, a_n\) if \(a_1 + a_3 + a_5 = -12\) and \(a_1 a_3 a_5 = 80\), then which of the following is/are true?</p>
Step-by-Step Solution
Key Concept: In an A.P., if we denote the middle term as the center, odd-positioned terms form a symmetric pattern: a₁, a₃, a₅ can be written as (a-2d), a, (a+2d). This symmetry allows us to use sum and product constraints to find individual terms.
<p><strong>Step 1:</strong> Express a₁, a₃, a₅ in terms of a₃ and common difference d.</p><p>Since a₃ is the middle term: a₁ = a₃ - 2d, a₅ = a₃ + 2d</p><p><strong>Step 2:</strong> Use the sum condition.</p><p>(a₃ - 2d) + a₃ + (a₃ + 2d) = -12</p><p>3a₃ = -12 ⟹ a₃ = -4</p><p><strong>Step 3:</strong> Use the product condition.</p><p>(a₃ - 2d) · a₃ · (a₃ + 2d) = 80</p><p>a₃(a₃² - 4d²) = 80</p><p>-4(16 - 4d²) = 80</p><p>-64 + 16d² = 80</p><p>16d² = 144 ⟹ d² = 9 ⟹ d = ±3</p><p><strong>Step 4:</strong> Apply the increasing A.P. constraint.</p><p>Since the A.P. is increasing, d > 0, so d = 3</p><p><strong>Step 5:</strong> Find the three terms.</p><p>a₁ = -4 - 6 = -10</p><p>a₃ = -4</p><p>a₅ = -4 + 6 = 2</p><p><strong>Verification:</strong> -10 + (-4) + 2 = -12 ✓ and (-10)(-4)(2) = 80 ✓</p><p>∴ Answer depends on options ACD (typical statements: a₁ = -10, d = 3, a₅ = 2, or related properties)</p>
Correct Answer: ACD