lf a, b, c are pth, qth and rth terms of a GP, then (q - r) log a + (r - p) log b + (p - q) log c is equal to

Let A and R be the first term and common ratio of the given GP. Then, a = AR^P-1

$\begin{array}{l}\Rightarrow \log \mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{R}-----\left(\mathrm{i}\right)\\ \mathrm{similarly}, \log \mathrm{b}=\log \mathrm{A}+(\mathrm{q}-1)\log \mathrm{R}----\left(\mathrm{ii}\right)\\ \mathrm{and} \log \mathrm{c}=\log \mathrm{A}+(\mathrm{r}-1)\log \mathrm{R}-----\left(\mathrm{iii}\right)\end{array}$

now , (q - r) log a + (r - p) log b + (p - q) log c

$\begin{array}{l}=(\mathrm{q}-\mathrm{r})\{\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{R}\}+(\mathrm{r}-\mathrm{p})\{\log \mathrm{A}+(\mathrm{q}-1)\log \mathrm{R}\}+(\mathrm{p}-\mathrm{q})\{\log \mathrm{A}+(\mathrm{r}-1)\log \mathrm{R}\}\\ \\ \\ =\log \mathrm{A}[\mathrm{q}-\mathrm{r}+\mathrm{r}-\mathrm{p}+\mathrm{p}-\mathrm{q}]+\log \mathrm{R}\left[\mathrm{p}\right(\mathrm{q}-\mathrm{r})+\mathrm{q}(\mathrm{r}-\mathrm{p})+\mathrm{r}(\mathrm{p}-\mathrm{q})-(\mathrm{q}-\mathrm{r})-(\mathrm{r}-\mathrm{p})-(\mathrm{p}-\mathrm{q}\left)\right]\\ \\ \\ =\log \mathrm{A}\cdot 0+\log \mathrm{R}\cdot 0=0\end{array}$