Example 3 In the positive term sequence $\left\{a_{n}\right\}$, $a_{1}=10, a_{k+1}$ $=10 \sqrt{a_{k}}$. Find the general term formula $a_{n}$.
(1991, Sichuan Province High School Mathematics Competition)
Solution: Let $x_{k}=\lg a_{k}$. Then
$$
x_{k+1}=1+\frac{1}{2} x_{k}, x_{k}=1+\frac{1}{2} x_{k-1} \text {. }
$$

From this, we get $x_{i+1}-x_{k}=\frac{1}{2}\left(x_{k}-x_{k-1}\right)$, and $x_{2}-x_{1}=\frac{1}{2}$.
Let $k$ take $1,2, \cdots, n$, the sum is
$$
\begin{array}{l}
x_{n+1}-x_{1}=1-\left(\frac{1}{2}\right)^{n}, \\
x_{n+1}=x_{1}+1-\left(\frac{1}{2}\right)^{n}=2-\left(\frac{1}{2}\right)^{n} .
\end{array}
$$

Thus, we have
$$
a_{n}=10^{2-\left(\frac{1}{2}\right)^{n}} \text {. }
$$