1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=2000$, if $f(0)=1$ and for any $x$ the equality $f(x+2)=f(x)+4 x+2$ holds.
Answer: 3998001

Solution: In the equation $f(x+2)-f(x)=4 x+2$, we will substitute for $x$ the numbers $0, 2, 4, \ldots, 1998$. We get:

$$
\begin{aligned}
& f(2)-f(0)=4 \cdot 0+2 \\
& f(4)-f(2)=4 \cdot 2+2
\end{aligned}
$$

$$
f(2000)-f(1998)=4 \cdot 1998+2
$$

Adding the equations, we get: $f(2000)-f(0)=4 \cdot(0+2+4+\cdots+1998)+2 \cdot 1000=$ $4 \cdot \frac{1998 \cdot 1000}{2}+2000=3998000$. Then $f(2000)=3998001$.