$a^{x}=b^{y}=c^{z}=d^{u t}$ Let, $a^{x}=b^{y}=c^{z}=d^{u t}=k$ $\Rightarrow a=k^{1 / x}, b=k^{1 / y}, c=k^{1 / z}, d=k^{1 / u}$...(i)
$a, b, c, d$ are in GP.
$$
\begin{array}{l}
\therefore \frac{b}{a}=\frac{c}{b}=\frac{d}{c} \\
\Rightarrow \frac{k^{1 / y}}{k^{1 / x}}=\frac{k^{1 / z}}{k^{1 / y}}=\frac{k^{1 / u}}{k^{1 / z}}\{\text { using Eq. (i) }\} \\
\Rightarrow k^{\frac{1}{y} \frac{1}{x}}=k^{\frac{1}{z} \frac{1}{y}}=\frac{1}{k^{u}} \frac{1}{z} \\
\Rightarrow \frac{1}{y}-\frac{1}{x}=\frac{1}{z}-\frac{1}{y}=\frac{1}{u}-\frac{1}{z}
\end{array}
$$
$\therefore \frac{1}{x}, \frac{1}{y}, \frac{1}{z}, \frac{1}{u}$ are in A.P.
$\Rightarrow x, y, z, u$ are in H.P.