The value of the determinant $\begin{vmatrix}bc &ca& ab\\p&q&r\\1&1&1\end{vmatrix}$, where a, b and c are respectively the pth, qth and rth terms of a H.P., is

0

Let D be the common difference of the corresponding A.P. and A be its first term. Then,

$\frac{1}{a}=A+(p-1)D,\frac{1}{b}=A+(q-1)D$ and $\frac{1}{c}=A+(r-1)D$

Now,

$\begin{vmatrix}bc &ca& ab\\p&q&r\\1&1&1\end{vmatrix}$

$=abc\begin{vmatrix}\frac{1}{a} &\frac{1}{b}& \frac{1}{c}\\p&q&r\\1&1&1\end{vmatrix}$ [Applying $R_1→ R_1(\frac{1}{abc})$]

$=abc\begin{vmatrix}\frac{1}{a} &\frac{1}{b}& \frac{1}{c}\\p-1&q-1&r-1\\1&1&1\end{vmatrix}$  [Applying $R_2 → R_2-R_3$]

$=abc\begin{vmatrix}\frac{1}{a}-(p-1)D &\frac{1}{b}-(q-1)D& \frac{1}{c}-(r-1)D\\p-1&q-1&r-1\\1&1&1\end{vmatrix}$  [Applying $R_1→ R_1-DR_2$]

$=abc\begin{vmatrix}A&A&A\\p-1&q-1&r-1\\1&1&1\end{vmatrix}=abc×0=0$