Task B-4.4. A sequence of real numbers is given by the formula $x_{n+1}=\frac{n+1}{x_{n}}$, for all $n \geq 1$, where $x_{1}=123456789$. What is $x_{1} \cdot x_{2} \cdot x_{18} \cdot x_{19} \cdot x_{52} \cdot x_{53}$?
## Solution.

Notice that from the given equality it follows that

$$
x_{n} \cdot x_{n+1}=n+1, \text { for all } n \geq 1
$$

Then, $x_{1} \cdot x_{2}=2, x_{18} \cdot x_{19}=19, x_{52} \cdot x_{53}=53$

so $x_{1} \cdot x_{2} \cdot x_{18} \cdot x_{19} \cdot x_{52} \cdot x_{53}=2 \cdot 19 \cdot 53=2014$.