1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=4000$, if $f(0)=1$ and for any $x$ the equality $f(x+4)=f(x)+3 x+4$ holds.
Answer: 5998001

Solution: In the equation $f(x+4)-f(x)=3 x+4$, we will substitute for $x$ the numbers $0,4,8, \ldots, 3996$. We get:

$$
\begin{aligned}
& f(4)-f(0)=3 \cdot 0+4 \\
& f(8)-f(4)=3 \cdot 4+4
\end{aligned}
$$

$$
f(4000)-f(3996)=3 \cdot 3996+4
$$

Adding the equations, we get: $f(4000)-f(0)=3 \cdot(0+4+8+\cdots+3996)+4 \cdot 1000=$ $3 \cdot \frac{3996 \cdot 1000}{2}+4000=5998000$. Then $f(4000)=5998001$.