Example 12 The sports meet lasted for $n$ days $(n>1)$, and a total of $m$ medals were awarded. On the first day, 1 medal was awarded, and then $\frac{1}{7}$ of the remaining $m-1$ medals were awarded. On the second day, 2 medals were awarded, and then $\frac{1}{7}$ of the remaining medals were awarded, and so on, until on the $n$-th day, exactly $n$ medals were awarded, and all were distributed. How many days did the sports meet last? How many medals were awarded in total?
(IMO - 9 problem)
Let $a_{k}$ be the number of medals awarded on the $k$-th day, then
$$
a_{1}=1+\frac{1}{7}(m-1)=\frac{1}{7}(m+6), a_{k}=k+\frac{1}{7}\left(m-a_{1}-a_{2}-\cdots-a_{k-1}-k\right) \text {, }
$$

Thus, $a_{k}-a_{k-1}=1+\frac{1}{7}\left(-a_{k-1}-1\right)$, which means $a_{k}=\frac{6}{7} a_{k-1}+\frac{6}{7}$.
Since the fixed point of $f(x)=\frac{6}{7} x+\frac{6}{7}$ is 6, we have
$$
\begin{array}{l}
a_{k}=\left(\frac{6}{7}\right)^{k-1}\left(a_{1}-6\right)+6=\frac{1}{7}\left(\frac{6}{7}\right)^{k-1}(m-36)+6, \\
m=a_{1}+a_{2}+\cdots+a_{n}=(m-36)\left[1-\left(\frac{6}{7}\right)^{n}\right]+6 n,
\end{array}
$$

Solving this, we get $m=\frac{7^{n}}{6^{n-1}}(n-6)+36$.
Since $m, n$ are positive integers and $7^{n}$ is coprime with $6^{n-1}$, it follows that $6^{n-1}$ divides $(n-6)$. Also, for $n>1$, it is clear that $|n-6|<6^{n-1}$, so the only solution is $n=6$, which gives $m=36$. Therefore, the sports meet lasted 6 days, and a total of 36 medals were awarded.