If ( a 0, ~b 0, c 0) are respectively the ( pth, q ) th, rth terms...

Let $A$ be the 1 st term and ($\mathrm{R}$) the common ratio of G.P., then \$($$\begin{aligned} \text{ and } a=T_p=A R^{p-1} \\ \text{ and } \therefore \log\mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1) \log\mathrm{R} \end{aligned}$) Similarly, (\log $\mathrm{b}=\log \mathrm{A}+(\mathrm{q}-$1) \log $\mathrm{R}$) and (\log $\mathrm{c}=\log \mathrm{A}+(\mathrm{r}-$1) \log $\mathrm{R}$) (\therefore $\Delta=\left|\begin{array}{lll} \logA+(p-1) \log R \text{ and } p \text{ and } 1 \\ \log A+(q-1) \log R \text{ and } q \text{ and } 1 \\ \log A+(r-1) \log R \text{ and } r \text{ and } 1 \end{array}\right|) Split into two determinants and in the first take (\log A) common and in the second take (\log $\mathrm{R}$) common (\Delta=\logA\left|\begin{array}{lll}1 \text{ and } p \text{ and } 1 \\ 1 \text{ and } q \text{ and } 1 \\ 1 \text{ and } r \text{ and } 1 \end{array}\right|+\log R\left|\begin{array}{lll}p-1 \text{ and } p \text{ and } 1 \\ q-1 \text{ and } q \text{ and } 1 \\ r-1 \text{ and } r \text{ and } 1 \end{array}\right|) Apply (\mathrm{C}}_{1} \rightarrow \mathrm{C}}_{2} \rightarrow \mathrm{C}}_{3}) in the second \$(\Delta=$0+$\log R\left|$\begin{array}{lll}0 & p & 1 \\ 0 & q & 1 \\ 0 & r & 1 \end{array}$$$\right|=0)