If a_1, a_2, a_3, dots, a_n form a geometric | vedclass

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(A) Let the common ratio of the geometric progression be $r$. Then $a_{n+k} = a_n \cdot r^k$.Taking the logarithm,we get $\log a_{n+k} = \log a_n + k

Vedclass · Accepted Answer

$0$

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(A) Let the common ratio of the geometric progression be $r$. Then $a_{n+k} = a_n \cdot r^k$.Taking the logarithm,we get $\log a_{n+k} = \log a_n + k \log r$.Let $D$ be the determinant.$D = \left| \begin{array}{ccc} \log a_n & \log a_n + \log r & \log a_n + 2\log r \ \log a_{n+3} & \log a_{n+3} + \log r & \log a_{n+3} + 2\log r \ \log a_{n+6} & \log a_{n+6} + \log r & \log a_{n+6} + 2\log r \end{array} ight|$.Applying the column operations $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_2$,we get:$D = \left| \begin{array}{ccc} \log a_n & \log r & \log r \ \log a_{n+3} & \log r & \log r \ \log a_{n+6} & \log r & \log r \end{array} ight|$.Since column $C_2$ and column $C_3$ are identical,the value of the determinant is $0$.

If $a_1, a_2, a_3, \dots, a_n$ form a geometric progression,find the value of the determinant: $\left| \begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array} \right|$.

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