Example 2. Let $y \neq-1$. We set,

$$
x_{1}=\frac{y-1}{y+1}, \quad x_{2}=\frac{x_{1}-1}{x_{1}+1}, \quad x_{3}=\frac{x_{2}-1}{x_{2}+1}, \ldots
$$

What is $y$ if $x_{1978}=-\frac{1}{3}$?
Solution. Substituting the value of $x_{1}$ into the second equality, after simplifications we get:

$$
x_{2}=-\frac{1}{y}
$$

Further,

$$
x_{3}=\frac{y+1}{1-y}, \quad x_{4}=y, \quad x_{5}=\frac{y-1}{y+1}=x_{1}
$$

Therefore,

$$
x_{5}=x_{1}, \quad x_{6}=x_{2}, \quad x_{7}=x_{3}, \quad x_{8}=x_{4}, \ldots, \quad x_{1978}=x_{2}=-\frac{1}{y}
$$

from which $y=3$.

## EXERCISES

174. Let a number $N$ be given and a sequence of positive numbers $x_{1}, x_{2}, \ldots, x_{n}$, where each subsequent term is formed by the rule

$$
x_{n}=\frac{1}{2}\left(x_{n-1}+\frac{N}{x_{n-1}}\right)
$$

Prove that

$$
\lim _{n \rightarrow \infty} x_{n}=\sqrt{N}
$$

175. A sequence is defined by the recurrence formula

$$
u_{n}=(\alpha+\beta) u_{n-1}-\alpha \beta u_{n-2}
$$

and initial values $u_{1}=\alpha+\beta, u_{2}=\frac{\alpha^{3}-\beta^{3}}{\alpha-\beta}$. Find the general term of the sequence.

176. Two sequences are given: $x_{0}, x_{1}, \ldots, x_{n}, \ldots\left(x_{0}>0\right)$ and $y_{0}, y_{1}, \ldots, y_{n}, \ldots\left(y_{0}>0\right)$, where

$$
x_{n}=\frac{x_{n-1}+y_{n-1}}{2}, \quad y_{n}=\sqrt{x_{n-1} y_{n-1}}
$$

Prove that the limits $\lim _{n \rightarrow \infty} x_{n}$ and $\lim _{n \rightarrow \infty} y_{n}$ exist and are equal to each other.

177. Two positive numbers $a$ and $b(a>b)$ and two sequences are given:

$$
\begin{array}{ll}
a_{1}=\frac{a+b}{2}, & b_{1}=\frac{2 a b}{a+b} \\
a_{2}=\frac{a_{1}+b_{1}}{2}, & b_{2}=\frac{2 a_{1} b_{1}}{a_{1}+b_{1}} \\
\cdots \cdot \cdot \cdot \cdot, & . \cdot \cdot \cdot \cdot \\
a_{n+1}=\frac{a_{n}+b_{n}}{2}, & b_{n+1}=\frac{2 a_{n} b_{n}}{a_{n}+b_{n}}
\end{array}
$$

Find the limits of these sequences.

178. Given 1979 numbers: $x_{1}=0, x_{2}, \ldots, x_{1978}, x_{1979}=0$. Moreover, $x_{i}=\frac{x_{i-1}+x_{i+1}}{4}+1$ for any $i=2,3, \ldots, 1978$. Prove that:

a) $0 \leqslant x_{i} \leqslant 2$ for $i=1,2, \ldots, 1979$

b) $x_{1}x_{992}>\ldots>x_{1979}$

c) $x_{1}=x_{1979}, x_{2}=x_{1978}, \ldots$.

179. Find the sum $\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{n}{(n+1)!}$ and its limit as $n \rightarrow \infty$.
180. Compute the limits:
a) $\lim \sqrt[n]{n}$;
b) $\lim _{n \rightarrow \infty} \sqrt[n]{\frac{1}{n!}}$
c) $\lim _{n \rightarrow \infty}\left(\frac{5}{9} \cdot \frac{14}{20} \cdot \frac{27}{35} \ldots \frac{2 n^{2}-n-1}{2 n^{2}+n-1}\right)$.
181. Prove that for $x \neq 0$ we have:

$$
\lim _{n \rightarrow \infty}\left(\cos \frac{x}{2} \cdot \cos \frac{x}{4} \ldots \cos \frac{x}{2^{n}}\right)=\frac{\sin x}{x}
$$

182. Find $\lim _{n \rightarrow \infty} x_{n}$, if:

58
a) $x_{n}=\sqrt{\underbrace{2+\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}_{n \text { times }}} ;$

b) $x_{n}=\sqrt{\underbrace{2 \sqrt{2 \sqrt{2 \ldots \sqrt{2}}}}_{n \text { times }}}$;

![](https://cdn.mathpix.com/cropped/2024_05_21_ee3d8c3d885a47ff5ccfg-60.jpg?height=234&width=694&top_left_y=591&top_left_x=410)

## § 4. Various Algebraic Problems