## Task $9 / 74$

The limit of the sequence $\left\{a_{n}\right\}$ is sought, which has the following properties:

1. $a_{n}=\frac{b_{n}}{c_{n}}$
2. $b_{1} ; c_{1}$ are natural numbers
3. $b_{n+1}=2 b_{n}$
4. $c_{n+1}=b_{n}+c_{n}$
It is

$$
a_{n+1}=\frac{b_{n+1}}{c_{n+1}}=\frac{2 b_{n}}{b_{n}+c_{n}}=\frac{b_{n}+c_{n}+b_{n}-c_{n}}{b_{n}+c_{n}}=1+\frac{b_{n}-c_{n}}{b_{n}+c_{n}}
$$

Due to properties 2. and 3., the sequence $\left\{b_{n}\right\}$ is monotonically increasing; due to properties 2., 3., and 4., the sequence $\left\{c_{n}\right\}$ is monotonically increasing, and thus the sequence $\left\{b_{n}+c_{n}\right\}$ is also monotonically increasing. From properties 3. and 4., it further follows that

$$
c_{n+1}-b_{n+1}=b_{n}+c_{n}-2 b_{n}=c_{n}-b_{n}
$$

Since this relationship holds for any $n$, it follows that $c_{n}-b_{n}=$ constant. Therefore,

$$
a_{n+1}=1+\frac{\text { constant }}{b_{n}+c_{n}} \quad \text { and } \quad \lim _{n \rightarrow \infty} a_{n+1}=\lim _{n \rightarrow \infty} a_{n}=1
$$