Example 6 There are two rivers, A and B, each with a flow rate of $300 \mathrm{~m}^{3} / \mathrm{s}$, which converge at a certain point and continuously mix. Their sediment contents are $2 \mathrm{~kg} / \mathrm{m}^{3}$ and $0.2 \mathrm{~kg} / \mathrm{m}^{3}$, respectively. Assume that from the convergence point, there are several observation points along the shore. During the flow between two adjacent observation points, the mixing effect of the two streams is equivalent to the exchange of $100 \mathrm{~m}^{3}$ of water between the two streams in 1 second, i.e., $100 \mathrm{~m}^{3}$ of water flows from stream A to stream B, mixes, and then $100 \mathrm{~m}^{3}$ of water flows from stream B back to stream A, and mixes together. Question: From which observation point onwards will the difference in sediment content between the two streams be less than $0.01 \mathrm{~kg} / \mathrm{m}^{3}$ (ignoring sedimentation)?
(1999, Comprehensive Test for University Admission of Outstanding Students)
Solution: Let the sand content of two water flows be $a \, \text{kg} / \text{m}^{3}$ and $b \, \text{kg} / \text{m}^{3}$, and the water volumes flowing through in a unit time be $p \, \text{m}^{3}$ and $q \, \text{m}^{3}$, respectively. Then the sand content after mixing is
$$
\frac{a p + b q}{p + q} \, \text{kg} / \text{m}^{3}.
$$

At the $n$-th observation point, the sand content of water flow $A$ is $a_{n} \, \text{kg} / \text{m}^{3}$, and the sand content of water flow $B$ is $b_{n} \, \text{kg} / \text{m}^{3}$, where $n = 1, 2, \cdots$. Then
$$
\begin{aligned}
a_{1} & = 2, b_{1} = 0.2, \\
b_{n} & = \frac{1}{400} \left(300 b_{n-1} + 100 a_{n-1}\right) \\
& = \frac{1}{4} \left(3 b_{n-1} + a_{n-1}\right). \\
a_{n} & = \frac{1}{300} \left(200 a_{n-1} + 100 b_{n}\right) \\
& = \frac{1}{4} \left(3 a_{n-1} + b_{n-1}\right).
\end{aligned}
$$

Thus, $a_{n} - b_{n} = \frac{1}{2} \left(a_{n-1} - b_{n-1}\right)$
$$
= \cdots = \frac{1}{2^{n-1}} \left(a_{1} - b_{1}\right) = \frac{1}{2^{n-1}}.
$$
$$
2^{n-1} > 180, \quad n \geqslant 9.
$$

From this, we can see that starting from the 9th observation point, the difference in sand content between the two water flows is less than $0.01 \, \text{kg} / \text{m}^{3}$.