If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in G.P., then the value of the determinant $ \left|\begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array} ight| ext {, is } $

Question

If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in G.P., then the value of the determinant $ \left|\begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array}ight| 	ext {, is } $

Aicharya · Accepted Answer

$ \begin{aligned} & \left|\begin{array}{lll} \log a_n & \log a_{n+1} & \log a_{n+2} \ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array}ight| \ & C_3 ightarrow C_3-C_2, C_2 ightarrow C_3-C_1 \ & =\left|\begin{array}{lll} \log a_n & \log r & \log r \ \log a_{n+3} & \log r & \log r \ \log a_{n+6} & \log r & \log r \end{array}ight|=0 \end{aligned} $ (where $r$ is a common ratio).

If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in G.P., then the value of the determinant $…

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