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Q.
For three positive integers $p , q , r , x^{p q^2}=y^{y r}=z^{p^2 r }$ and $r = pq +1$ such that $3,3 \log _y x, 3 \log _z y$ $7 \log _x z$ are in A.P. with common difference $\frac{1}{2}$. Then $r-p-q$ is equal to
$ pq ^2=\log _{ x } \lambda $
$ qr =\log _{ y } \lambda $
$p ^2 r =\log _{ z } \lambda$
$ \log _{ y } x =\frac{ qr }{ pq ^2}=\frac{ r }{ pq } \ldots \ldots .(1) $
$ \log _{ x } z =\frac{ pq ^2}{ p ^2 r }=\frac{ q ^2}{ pr } \ldots \ldots \ldots(2) $
$ \log _{ z } y =\frac{ p ^2 r }{ qr }=\frac{ p ^2}{ q } \ldots \ldots \ldots(3)$
$3, \frac{3 r }{ pq }, \frac{3 p ^2}{ q }, \frac{7 q ^2}{ pr } \text { in A.P } $
$\frac{3 r }{ pq }-3=\frac{1}{2}$
$ r =\frac{7}{6} pq\ldots \ldots \ldots(4)$
$ r = pq +1 $
$ pq =6\ldots \ldots \ldots(5)$
$ r =7 \ldots\ldots \ldots \ldots(6)$
$\frac{3 p ^2}{ q }=4$
After solving $p=2$ and $q=3$