If $a, b, c$ are in A.P. $p, q, r$ are in H.P. and $ap, bq, cr$ are in G.P., then$\frac{p}{r}+\frac{r}{p}$ is equal to

According to the given condition
   $2b=a+c,q=\frac{2pr}{p+r}$ and $b^2{q}^2=\left(ap\right)\left(cr\right)$
Substituting values of $b$ and $q$ from first two expressions in the last expression, we get
$\begin{array}{l} \frac{(a+c{)}^2}{4}{\left(\frac{2pr}{p+r}\right)}^2=\left(ac\right)\left(pr\right)\\ \Rightarrow \frac{(a+c{)}^2}{ac}=\frac{(p+r{)}^2}{pr} \Rightarrow \frac{a}{c}+\frac{c}{a}=\frac{p}{r}+\frac{r}{p}\end{array}$