Question5p 2. Fie expresia $E(x)=\left(\frac{1}{x-3}+\frac{x}{9-x^{2}}\right): \frac{3}{(x-3)(x+3)}$, unde $x \in \mathbb{R} \backslash\{ \pm 3\}$. (2p) a) Arată că $E(-1)=1$. (3p) b) Calculează numărul $z=E(1)+2 E(2)+3 E(3)+\ldots+2022 E(2022)$.

Studdy Solution
Calculate the sum $z = E(1) + 2E(2) + 3E(3) + \ldots + 2022E(2022)$.
Since $E(x) = 1$ for all $x \neq \pm 3$, each term in the sum is simply the coefficient:
$$z = 1 \times 1 + 2 \times 1 + 3 \times 1 + \ldots + 2022 \times 1$$
This is the sum of the first 2022 natural numbers:
$$z = 1 + 2 + 3 + \ldots + 2022$$
The formula for the sum of the first $n$ natural numbers is:
$$\frac{n(n+1)}{2}$$
Substituting $n = 2022$:
$$z = \frac{2022 \times 2023}{2}$$
Calculate the product:
$$z = \frac{2022 \times 2023}{2} = 2047173$$
The value of $z$ is $\boxed{2047173}$.