Explanation

(c)
Given ,$a_{1},a_{2},\ a_{3},\ldots \in$GP
$\Rightarrow \log{a_{1},\log{a_{2},\ldots \in}AP}$
$\Rightarrow \log{a_{n},\log{a_{n + 1},\log{a_{n + 2},\ldots \in}}}$AP
$\Rightarrow \log{a_{n + 1} = \frac{\log{a_{n} + \log a_{n + 2}}}{2}}$ …(i)
Similarly, $\log{a_{n + 4} =}\frac{\log{a_{n + 3} + \log a_{n + 5}}}{2}$ …(ii)
and $\log{a_{n + 7} = \frac{\log{a_{n + 6} + \log a_{n + 8}}}{2}}$ …(ii)
Given, $\mathrm{\Delta} = \left| \begin{matrix}
\log a_{n} & \log a_{n + 1} & \log a_{n + 2} \\
\log a_{n + 3} & \log a_{n + 4} & \log a_{n + 5} \\
\log a_{n + 6} & \log a_{n + 7} & \log a_{n + 8}
\end{matrix} \right|$
Applying $C_{2} \rightarrow C_{2} - \frac{C_{1} + C_{3}}{2}$
$\mathrm{\Delta} = \left| \begin{matrix}
\log a_{n} & 0 & \log a_{n + 2} \\
\log a_{n + 3} & 0 & \log a_{n + 5} \\
\log a_{n + 6} & 0 & \log a_{n + 8}
\end{matrix} \right| = 0$