Explanation

(d)
Let $A$ and $R$ be the first term and common ratio respectively.
$\therefore l = AR^{p - 1}$
$\Rightarrow \log{l = \log{A + (p - 1)\log R}}$
$m = AR^{q - 1}$
$\Rightarrow \log{m = \log{A + (q - 1)\log R}}$
and $n = AR^{r - 1}$
$\Rightarrow$log $n = \log{A + (r - 1)\log R}$
Now, $\left| \begin{matrix}
\log l & p & 1 \\
\log m & q & 1 \\
\log n & r & 1
\end{matrix} \right| = \left| \begin{matrix}
\log{A + (p - 1)\log R} & p & 1 \\
\log{A + (q - 1)\log R} & q & 1 \\
\log{A + (r - 1)\log R} & r & 1
\end{matrix} \right|$
On multiplying $R_{1},R_{2}$and$\ R_{3}$by$\ (q - r),(r - p)$and$\ (p - q)$and adding$\ R_{1} + R_{2} + R_{3},\$we get
$= (q - r + r - p + p - q).\log{A + \{(q - r)(p - 1)}$
$+ (r - p)(q - 1) + (p - q)(r - 1)\}\ log\ R$
=0