A businessman took out a subsidized loan of 12 million Ft at an annual fixed interest rate of $8 \%$. What will his debt be after 10 years if he can pay off 1.2 million forints annually?
Solution. At the end of the first year, the debt will be $12 \cdot 1.08 - 1.2$ million forints; at the end of the second year, $(12 \cdot 1.08 - 1.2) \cdot 1.08 - 1.2$, and so on. At the end of the 10th year, the debt (in million forints) is:

$$
\begin{aligned}
x & =(((12 \cdot 1.08 - 1.2) \cdot 1.08 - 1.2) \cdot 1.08 \ldots) \cdot 1.08 - 1.2= \\
& =12 \cdot 1.08^{10} - 1.2 \cdot\left(1 + 1.08 + 1.08^{2} + \ldots + 1.08^{9}\right)
\end{aligned}
$$

In the parentheses, the first 10 terms of a geometric sequence are present, with the common ratio being 1.08. Using the known summation formula, we get:

$$
x = 12 \cdot 1.08^{10} - 1.2 \cdot \frac{1.08^{10} - 1}{1.08 - 1}
$$

from which $x \approx 8.523225$.

Thus, at the end of the 10th year, the entrepreneur's debt is $8523225 \mathrm{Ft}$.