If \(p\left(\frac{1}{q}+\frac{1}{r}\right), q\left(\frac{1}{r}+\frac{1}{p}\right), r\left(\frac{1}{p}+\frac{1}{q}\right)\) are in \(\mathrm{AP}\), then \(p, q, r\)

\(p\left(\frac{1}{q}+\frac{1}{r}\right) q\left(\frac{1}{r}+\frac{1}{p}\right) r\left(\frac{1}{p}+\frac{1}{q}\right)\)
\(\begin{aligned} & =p\left(\frac{q+r}{q r}\right), q\left(\frac{p+r}{p r}\right), r\left(\frac{p+q}{p q}\right) \\ & =\frac{p(q+r)}{q r}+1, \frac{q(p+r)}{p r}+1, r\left(\frac{p+q}{p q}\right)+1\end{aligned}\)
(On adding \(l\) each term)
\(=\frac{p q+p r+q r}{q r}, \frac{q p+q r+p r}{p r}, \frac{r p+r q+p q}{p q}\)
\(\begin{aligned} & =\frac{1}{q r}, \frac{1}{p r}, \frac{1}{p q} \text { in AP } \\ & =\frac{p q r}{q r}, \frac{p q r}{p r}, \frac{p q r}{p q} \text { in APs }\end{aligned}\)
(on multiply pqr both order)
\(\Rightarrow p, q, r\) also in AP.