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The given determinant=
\(\(\begin{gathered}\sum {r\log b - q\log c} =\sum {\log \frac{{{b^r}}}{{{c^q}}}}\hfill \\ Let,a = A{R^{p - 1}},b = A{R^{q - 1}},c = A{R^{r - 1}}\hfill \\\therefore {b^r} = {A^r}{R^{rq - r}},{c^q} = {A^q}{R^{qr - q}}\hfill \\\therefore \frac{{{b^r}}}{{{c^q}}} = {A^{r - q}}{R^{q - r}} = {\left( {\frac{A}{R}} \right)^{r - q}} \hfill \\Now,\sum {\log \frac{{{b^r}}}{{{c^q}}}}\hfill \\= \sum {(r - q)\log \left( {\frac{A}{R}} \right)}\hfill \\= \log \left( {\frac{A}{R}} \right)\sum {(r - q)}\hfill \\=log\left( {\frac{A}{R}} \right)\sum {(r - q) + (p - r) + (q - p) = 0}\hfill \\\end{gathered}\)\)