If $a_{n}(>0)$ be the $n^{\text {th }}$ term of a G.P. then $\left|\begin{array}{ccc}\log a_{n} & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8}\end{array}\right|$ is equal to

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If $a_{n}(>0)$ be the $n^{\text {th }}$ term of a G.P. then $\left|\begin{array}{ccc}\log a_{n} & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8}\end{array}\right|$ is equal to

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kritika · Answer 1 · 2021-12-26T04:58:25+0000

Ans: (D)
Hint : $\log a_{n}=\log \left(a r^{n-1}\right)=\log a+(n-1) \log r$
$$
\begin{array}{r}
\therefore\left|\begin{array}{lc}
\log \mathrm{a}+(\mathrm{n}-1) \log \mathrm{r} & \log \mathrm{a}+\mathrm{nlogr} \quad \log \mathrm{a}+(\mathrm{n}+1) \operatorname{logr} \\
\log \mathrm{a}+(\mathrm{n}+2) \log r & \log \mathrm{a}+(\mathrm{n}+3) \operatorname{logr} \log \mathrm{a}+(\mathrm{n}+4) \log \mathrm{r} \\
\log \mathrm{a}+(\mathrm{n}+5) \log \mathrm{r} & \log \mathrm{a}+(\mathrm{n}+6) \operatorname{logr} \log \mathrm{a}+(\mathrm{n}+7) \operatorname{logr}
\end{array}\right| \\
\downarrow \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}, \mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}
\end{array}
$$
$=\left|\begin{array}{ccc}\log a+(n-1) \log r & \log a+n \log r & \log a+(n+1) \log r \\ 3 \log r & 3 \log r & 3 \log r \\ 6 \log r & 6 \log r & 6 \log r\end{array}\right|$

If \(a_{n}(>0)\) be the \(n^{\text {th }}\) term of a G.P. then \(\left|\begin{array}{ccc}\log a_{n} & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8}\end{array}\right|\) is equal to

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