1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=4500$, if $f(0)=1$ and for any $x$ the equality $f(x+3)=f(x)+2 x+3$ holds.
Answer: 6750001

Solution: In the equation $f(x+3)-f(x)=2 x+3$, we will substitute for $x$ the numbers $0,3,6, \ldots, 4497$. We get:

$$
\begin{aligned}
& f(3)-f(0)=2 \cdot 0+3 \\
& f(6)-f(3)=2 \cdot 3+3
\end{aligned}
$$

$$
f(4500)-f(4497)=2 \cdot 4497+3
$$

Adding the equations, we get: $f(4500)-f(0)=2 \cdot(0+3+6+\cdots+4497)+3 \cdot 1500=$ $2 \cdot \frac{4497 \cdot 1500}{2}+4500=6750000$. Then $f(4500)=6750001$.