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(D) Given that ${a_1}, {a_2}, {a_3}, \dots$ are in $G.P.$ with common ratio $r$.Thus,${a_{n+1}} = {a_n} \cdot r$,which implies $\log {a_{n+1}} = \log

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(D) Given that ${a_1}, {a_2}, {a_3}, \dots$ are in $G.P.$ with common ratio $r$.Thus,${a_{n+1}} = {a_n} \cdot r$,which implies $\log {a_{n+1}} = \log {a_n} + \log r$.Similarly,$\log {a_{n+k}} = \log {a_n} + k \log r$.Let the determinant be $\Delta = \left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} & \log {a_{n+2}} \ \log {a_{n+3}} & \log {a_{n+4}} & \log {a_{n+5}} \ \log {a_{n+6}} & \log {a_{n+7}} & \log {a_{n+8}} \end{array} ight|$.Applying column operations ${C_2} 	o {C_2} - {C_1}$ and ${C_3} 	o {C_3} - {C_2}$:$\Delta = \left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} - \log {a_n} & \log {a_{n+2}} - \log {a_{n+1}} \ \log {a_{n+3}} & \log {a_{n+4}} - \log {a_{n+3}} & \log {a_{n+5}} - \log {a_{n+4}} \ \log {a_{n+6}} & \log {a_{n+7}} - \log {a_{n+6}} & \log {a_{n+8}} - \log {a_{n+7}} \end{array} ight|$.Since $\log {a_{n+k}} - \log {a_{n+k-1}} = \log r$ for any $k$,the determinant becomes:$\Delta = \left| \begin{array}{ccc} \log {a_n} & \log r & \log r \ \log {a_{n+3}} & \log r & \log r \ \log {a_{n+6}} & \log r & \log r \end{array} ight|$.Since column $2$ and column $3$ are identical,the value of the determinant is $0$.

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

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