Example 4 In the positive term sequence $\left\{a_{n}\right\}$, $a_{1}=10, a_{n+1}$ $=10 \sqrt{a_{n}}$. Find the general formula for this sequence.
Solution: Given $a_{1}>0, a_{n+1}=10 \sqrt{a_{n}}$, taking the logarithm on both sides, we get
$\lg a_{n+1}=1+\frac{1}{2} \lg a_{n}$.
Let $x_{n}=\lg a_{n}$, then we have
$$
x_{n+1}=\frac{1}{2} x_{n}+1, x_{n}=\frac{1}{2} x_{n-1}+1 \text {. }
$$

Subtracting the two equations, we get $x_{n+1}-x_{n}=\frac{1}{2}\left(x_{n}-x_{n-1}\right)$, and $x_{2}-x_{1}=\frac{1}{2}$. Therefore,
$$
\begin{array}{l}
x_{n+1}=x_{1}+\sum_{k=1}^{n}\left(x_{k+1}-x_{k}\right) \\
=1+1-\left(\frac{1}{2}\right)^{n}=2-\left(\frac{1}{2}\right)^{n} . \\
\text { Hence } a_{n}=10^{2-\left(\frac{1}{2}\right)^{n-1}} \text {. }
\end{array}
$$

Therefore, $a_{n}=10^{2-}\left(\frac{1}{2}\right)^{n-1}$.