Example 1. A person earns a monthly salary of 200 yuan and receives a bonus of 20 yuan at the end of the month. After receiving the salary each month, they arrange the monthly expenses according to $\frac{4}{5}$ of the current savings. How much will be the remaining balance in the twelfth month?
Let the balance at the end of the $n$-th month be $a_{\mathrm{n}}$ yuan. According to the problem,
$$
\begin{array}{l}
a_{1}=200 \times\left(1-\frac{4}{5}\right)+20=60, \\
a_{2}=260 \times\left(1-\frac{4}{5}\right)+20=72 .
\end{array}
$$

In general, $a_{0}=\left(a_{0-1}+200\right) \times\left(1-\frac{4}{5}\right)+20$, which means $a_{\mathrm{n}}=\frac{1}{5} a_{n-1}+60$.
Let $a_{1}=60, \quad a_{n}=\frac{1}{5} a_{1-1}+60$
From this, we get
$$
a_{n}=75-15\left(\frac{1}{5}\right)^{n-1} \text {. }
$$

Therefore, $a_{12}=75-15\left(\frac{1}{5}\right)^{11} \approx 75$ (yuan).
The balance at the end of the 12th month is approximately 75.