Example 2 In the positive term sequence $\left\{a_{n}\right\}$,
$$
a_{1}=10, a_{k+1}=10 \sqrt{a_{k}}(k=1,2, \cdots) \text {. }
$$

Find the general term formula $a_{n}$.
Solution: Let $b_{k}=\lg a_{k}$. Then $b_{k+1}=\frac{1}{2} b_{k}+1$.
Let $x=\frac{1}{2} x+1$, we get $x=2$. Then $b_{k+1}-2=\frac{1}{2}\left(b_{k}-2\right)$.
Therefore, the sequence $\left\{b_{n}-2\right\}$ is a geometric sequence with the first term $b_{1}-2=-1$ and the common ratio $\frac{1}{2}$.
Thus, $b_{n}-2=-1 \times\left(\frac{1}{2}\right)^{n-1}$.
Since $b_{n}=\lg a_{n}$, we have $a_{n}=10^{b}=10^{-\left(\frac{1}{2}\right)^{n-1}+2}$, i.e., $a_{n}=10^{2-}\left(\frac{1}{2}\right)^{n-1}\left(n \in \mathbf{N}_{+}\right)$.