If \(x, y, z\) are positive real numbers such that \(x^{12} = y^{16} = z^{24}\) and the three quantities \(3 \log_y x, 4 \log_z y, n \log_x z\) are in arithmetic progression, then the value of \(n\)…

The value of $n$ is $16$. The equal powers give $\log_y x=\frac43$, $\log_z y=\frac32$, and $\log_x z=\frac12$, so the progression begins $4,6$ and has last term $8$. The value $8$ is the whole term $\frac n2$.