212. How can one calculate, without tables, using the binomial theorem, with a certain approximation, the final capital formed from an initial capital $K$, lent at a compound interest rate of $p \%$ annually for $n$ years?
212. $\boldsymbol{K}_{n}=\boldsymbol{K}\left(1+\frac{p}{100}\right)^{n}$. Applying the binomial formula of Newton, we have

$$
\begin{aligned}
K_{n} & =K\left[1+n \cdot \frac{p}{100}+\frac{n(n-1)}{1 \cdot 2}\left(\frac{p}{100}\right)^{2}+\right. \\
& \left.+\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3}\left(\frac{p}{100}\right)^{3}+\cdots\right]
\end{aligned}
$$

In this expansion, the successive terms of which decrease very rapidly, we can limit ourselves to the first few terms depending on the desired accuracy. For example, for $p \leqslant 5$ and $n \geqslant 10$, the first 4 terms of the expansion already give a very small error, of course, relative, depending on the value of $K$.