If x, y, z are positive and the ${\mathrm{p}}^{\mathrm{th}},{\mathrm{q}}^{\mathrm{th}},{\mathrm{r}}^{\text{th }}$ terms respectively of a G.P., then $(\mathrm{q}-\mathrm{r})\log \mathrm{x}+(\mathrm{r}-\mathrm{p})\log \mathrm{y}+(\mathrm{p}-\mathrm{q})\log \mathrm{z}=$

$\mathrm{x}={\mathrm{aR}}^{\mathrm{p}-1}\Rightarrow \log \mathrm{x}=\log \mathrm{a}+(\mathrm{p}-1)\mathrm{R}$
Similarly logy and logz
$\begin{array}{l}\Rightarrow (\mathrm{q}-\mathrm{r})\log \mathrm{x}+(\mathrm{r}-\mathrm{p})\log \mathrm{y}+(\mathrm{p}-\mathrm{q})\log \mathrm{z}\\ =\log \mathrm{a}(\mathrm{q}-\mathrm{r}+\mathrm{r}-\mathrm{p}+\mathrm{q}-\mathrm{q})+\log \mathrm{R}(\mathrm{p}-1)(\mathrm{q}-\mathrm{r})\\ +(\mathrm{q}-1)(\mathrm{r}-\mathrm{p})+(\mathrm{r}-1)(\mathrm{p}-\mathrm{q})=0\end{array}$