ધારો કે ${{\text{x}}_{\text{1}}}\,{\text{,}}\,\,{{\text{x}}_{\text{2}}}\,{\text{,}}\,…….\,\,{{\text{x}}_{{{\text{n}}_{\text{1}}}}}$ અને  ${{\text{y}}_{\text{1}}}\,{\text{,}}\,\,{{\text{y}}_{\text{2}}}\,{\text{,}}\,\,……\,\,{{\text{y}}_{{{\text{n}}_{\text{2}}}}}$ એ અનુક્રમે ${n_1}$ અને ${{\text{n}}_{\text{2}}}$ કદ ની શ્રેણી છે. 

${{\text{G}}_{\text{1}}}\, = \,\,{({x_1}\,\, \times \,\,{x_2}\, \times \,\,…….\,\, \times \,\,{x_{{n_1}}})^{1/{n_1}}}\,……..\,\,(1)$

${G_2}\,\, = \,\,{({y_1}\,\, \times \,\,{y_2}\, \times \,\,…….\,\, \times \,\,{y_{{n_2}}})^{1/{n_2}}}\,……..\,\,(2)$

અને $G\,\, = \,\,{[({x_1}\,\, \times \,\,{x_2}\, \times \,\,……\,\, \times \,\,{x_{{n_1}}})\, \times \,({y_1}\,\, \times \,\,{y_2}\, \times \,\,….\,\,{y_{{n_2}}})]^{\frac{1}{{{n_1}\, + \,\,{n_2}}}}}$

$G\,\, = \,\,{(G_1^{{n_1}}\,\, \times \,\,G_2^{{n_2}})^{1/{n_1}\, + \,{n_2}}}\,\,\,\,\,[(1)\,\,\& \,\,(2)\,$ પરથી $]$

$\therefore \,\,\,\log \,\,G\,\, = \,\,\frac{{{n_1}\,\log \,{G_1}\,\, + \,\,{n_2}\,\log \,{G_2}}}{{{n_1}\, + \,\,{n_2}}}$