Progressions are not as easy as you think

With $r$ being the geometric ratio, we can write $x,y,z$ as $\dfrac{y}{r}, y, ry.$

With $x,2y,3z$ being in A.P., we have that

$2y = \dfrac{x + 3z}{2} \Longrightarrow 4y = \dfrac{y}{r} + 3ry.$

Now assuming that $y \ne 0$ we then have that

$4r = 1 +3r^{2} \Longrightarrow 3r^{2} - 4r + 1 = 0 \Longrightarrow (3r - 1)(r - 1) = 0.$

Now we could have $r = 1$ , in which case $x = y = z$ and $x,2y,3z$ will be arithmetic, but assuming that $x,y,z$ must be distinct then we end up with $r = \boxed{\dfrac{1}{3}}.$