$\text{ If }{\mathrm{a}}_1,{\mathrm{a}}_2,\dots ..,{\mathrm{a}}_{\mathrm{n}},\dots .\text{ form }\text{ a }\text{ G.P. }\text{ And }{\mathrm{a}}_{\mathrm{i}}>0,\text{ for }\text{ all }\mathrm{i}\ge 1\text{, }$
$\text{ then }\left|\begin{array}{l}\log {\mathrm{a}}_{\mathrm{n}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\log {\mathrm{a}}_{\mathrm{n}+1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\log {\mathrm{a}}_{\mathrm{n}+2}\\ \log {\mathrm{a}}_{\mathrm{n}+3}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\log {\mathrm{a}}_{\mathrm{n}+4}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\log {\mathrm{a}}_{\mathrm{n}+5}\\ \log {\mathrm{a}}_{\mathrm{n}+6}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\log {\mathrm{a}}_{\mathrm{n}+7}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\log {\mathrm{a}}_{\mathrm{n}+8}\end{array}\right|\text{ is equal to }$

$\text{ We have, }\Rightarrow {\mathrm{a}}_{\mathrm{n}+1}^2={\mathrm{a}}_{\mathrm{n}}{\mathrm{a}}_{\mathrm{n}+2}$
$\Rightarrow 2\log {\mathrm{a}}_{\mathrm{n}+1}=\log {\mathrm{a}}_{\mathrm{n}}+\log {\mathrm{a}}_{\mathrm{n}+2}$
Similarly
$\begin{array}{r}2\log {\mathrm{a}}_{\mathrm{n}+4}=\log {\mathrm{a}}_{\mathrm{n}+3}+\log {\mathrm{a}}_{\mathrm{n}+5}\\ 2\log {\mathrm{a}}_{\mathrm{n}+7}=\log {\mathrm{a}}_{\mathrm{n}+6}+\log {\mathrm{a}}_{\mathrm{n}+8}\end{array}$
Substituting these values in second column of determinant, we get
$\begin{array}{l}\mathrm{\Delta }=\frac{1}{2}\left|\begin{array}{ccc}\log {\mathrm{a}}_{\mathrm{n}}& \log {\mathrm{a}}_{\mathrm{n}}+\log {\mathrm{a}}_{\mathrm{n}+2}& \log {\mathrm{a}}_{\mathrm{n}+2}\\ \log {\mathrm{a}}_{\mathrm{n}+3}& \log {\mathrm{a}}_{\mathrm{n}+3}+\log {\mathrm{a}}_{\mathrm{n}+5}& \log {\mathrm{a}}_{\mathrm{n}+5}\\ \log {\mathrm{a}}_{\mathrm{n}+6}& \log {\mathrm{a}}_{\mathrm{n}+6}+\log {\mathrm{a}}_{\mathrm{n}+8}& \log {\mathrm{a}}_{\mathrm{n}+8}\end{array}\right|\\ \left.=\frac{1}{2}\left(0\right)=0\text{ using }{\mathrm{C}}_2\to {\mathrm{C}}_2-{\mathrm{C}}_1-{\mathrm{C}}_3\right\}\end{array}$