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If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in G.P., then the value of the determinant
$$
\left|\begin{array}{ccc}
\log a_n & \log a_{n+1} & \log a_{n+2} \\
\log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\
\log a_{n+6} & \log a_{n+7} & \log a_{n+8}
\end{array}\right| \text {, is }
$$
Options:
$$
\left|\begin{array}{ccc}
\log a_n & \log a_{n+1} & \log a_{n+2} \\
\log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\
\log a_{n+6} & \log a_{n+7} & \log a_{n+8}
\end{array}\right| \text {, is }
$$
Solution:
2903 Upvotes
Verified Answer
The correct answer is:
0
0
$$
\begin{aligned}
& \left|\begin{array}{lll}
\log a_n & \log a_{n+1} & \log a_{n+2} \\
\log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\
\log a_{n+6} & \log a_{n+7} & \log a_{n+8}
\end{array}\right| \\
& C_3 \rightarrow C_3-C_2, C_2 \rightarrow C_3-C_1 \\
& =\left|\begin{array}{lll}
\log a_n & \log r & \log r \\
\log a_{n+3} & \log r & \log r \\
\log a_{n+6} & \log r & \log r
\end{array}\right|=0
\end{aligned}
$$
(where $r$ is a common ratio).
\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}\right| \\
& C_3 \rightarrow C_3-C_2, C_2 \rightarrow 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}\right|=0
\end{aligned}
$$
(where $r$ is a common ratio).
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