<p>If \(a_1, a_2, a_3, \ldots, a_n, \ldots\) are in GP, then \(\Delta = \begin{vmatrix} \log a_n & \log(a_n r) & \log(a_n r^2) \\ \log(a_n r^3) & \log(a_n r^4) & \log(a_n r^5) \\ \log(a_n r^6) & \log(a_n r^7) & \log(a_n r^8) \end{vmatrix}\) equals:</p>
Step-by-Step Solution
Key Concept: Convert logarithms of GP terms using log properties: log(a_n r^k) = log(a_n) + k·log(r), then factor out common terms from rows/columns to reveal linear dependence among rows.
<p><strong>Step 1:</strong> Use logarithm property: log(a_n r^k) = log(a_n) + k·log(r). Let log(a_n) = a and log(r) = b.</p><p><strong>Step 2:</strong> Rewrite the determinant:</p><p>Δ = |a, a+b, a+2b|</p><p> |a+3b, a+4b, a+5b|</p><p> |a+6b, a+7b, a+8b|</p><p><strong>Step 3:</strong> Express each element as a + (multiple of b). Notice each row can be written in the form: [a + kb, a + (k+1)b, a + (k+2)b]</p><p><strong>Step 4:</strong> Perform column operations: C₂ → C₂ - C₁ and C₃ → C₃ - C₁:</p><p>Δ = |a, b, 2b|</p><p> |a+3b, b, 2b|</p><p> |a+6b, b, 2b|</p><p><strong>Step 5:</strong> Columns 2 and 3 are proportional (C₃ = 2C₂). When two columns are proportional, the determinant is zero.</p><p>∴ Answer: A (Δ = 0)</p>
Correct Answer: A