${{\text{a}}_{\text{1}}}{\text{, }}{{\text{a}}_{\text{2}}}{\text{, }}{{\text{a}}_{\text{3}}}{\text{ }}…………{\text{ , }}{{\text{a}}_{\text{n}}}$  સમગુણોત્તર શ્રેણીમાં છે  ધારો કે સામાન્ય ગુણોત્તર $r$ છે.

$D\,\, = \,\,\left| {\begin{array}{*{20}{c}}
  {\log \,{a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}} \\ 
  {\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}} \\ 
  {\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}} 
\end{array}} \right|$

$ = \,\,\left| {\begin{array}{*{20}{c}}
  {\log {a_n}}&{\log {a_n} + \log r}&{\log {a_n} + 2\log r} \\ 
  {\log {a_{n + 3}}}&{\log {a_{n + 3}} + \log r}&{\log {a_{n + 3}} + 2\log r} \\ 
  {\log {a_{n + 6}}}&{\log {a_{n + 6}} + \log r}&{\log {a_{n + 6}} + 2\log r} 
\end{array}} \right|$

$ = \,\,\left| {\begin{array}{*{20}{c}}
  {\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\,\,$

$\,(\,\,\because \,\,\,{c_2}\,\, = \,\,{c_3}\,\,)$