# Problem 4

Let $f(x)=x^{2}+3 x+2$. Calculate the expression

$\left(1-\frac{2}{f(1)}\right) \cdot\left(1-\frac{2}{f(2)}\right) \cdot\left(1-\frac{2}{f(3)}\right) \cdots\left(1-\frac{2}{f(2019)}\right)$.
Answer: $\frac{337}{1010}$

## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_bce66f37762fe9ce3bacg-3.jpg?height=60&width=1285&top_left_y=1740&top_left_x=288)

$$
\left(1-\frac{2}{f(n)}\right)=\frac{f(n)-2}{f(n)}=\frac{n^{2}+3 n}{n^{2}+3 n+2}=\frac{n(n+3)}{(n+1)(n+2)}
$$

Substituting this fraction for $\mathrm{n}=1,2, \ldots, 2019$ in the product and performing the cancellations, we get

$\frac{1 \cdot 4}{2 \cdot 3} \cdot \frac{2 \cdot 5}{3 \cdot 4} \cdot \frac{3 \cdot 6}{4 \cdot 5} \cdot \frac{4 \cdot 7}{5 \cdot 6} \cdots \cdot \frac{2018 \cdot 2021}{2019 \cdot 2020} \cdot \frac{2019 \cdot 2022}{2020 \cdot 2021}=\frac{1 \cdot 2022}{3 \cdot 2020}=\frac{337}{1010}$.