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

STEP 1

1. The expression $E(x)$ is defined as $E(x) = \left(\frac{1}{x-3} + \frac{x}{9-x^2}\right) : \frac{3}{(x-3)(x+3)}$ .
2. The domain of $E(x)$ excludes $x = \pm 3$ due to division by zero.
3. We need to simplify $E(x)$ and evaluate it at specific points.

STEP 2

1. Simplify the expression $E(x)$ .
2. Calculate $E(-1)$ and verify it equals 1.
3. Calculate the sum $z = E(1) + 2E(2) + 3E(3) + \ldots + 2022E(2022)$ .

STEP 3

Simplify the expression $E(x)$ .
First, consider the expression:
$E(x) = \left(\frac{1}{x-3} + \frac{x}{9-x^2}\right) : \frac{3}{(x-3)(x+3)}$
The division by a fraction is equivalent to multiplication by its reciprocal:
$E(x) = \left(\frac{1}{x-3} + \frac{x}{9-x^2}\right) \times \frac{(x-3)(x+3)}{3}$
Simplify the terms inside the parentheses:
$\frac{x}{9-x^2} = \frac{x}{-(x-3)(x+3)} = -\frac{x}{(x-3)(x+3)}$
Thus, the expression becomes:
$E(x) = \left(\frac{1}{x-3} - \frac{x}{(x-3)(x+3)}\right) \times \frac{(x-3)(x+3)}{3}$
Combine the fractions:
$\frac{1}{x-3} - \frac{x}{(x-3)(x+3)} = \frac{(x+3) - x}{(x-3)(x+3)} = \frac{3}{(x-3)(x+3)}$
Substitute back into the expression:
$E(x) = \frac{3}{(x-3)(x+3)} \times \frac{(x-3)(x+3)}{3} = 1$

STEP 4

Calculate $E(-1)$ and verify it equals 1.
Since we have simplified $E(x)$ to 1 for all $x \neq \pm 3$ , we have:
$E(-1) = 1$
Thus, $E(-1) = 1$ is verified.

STEP 5

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 $\boxed{2047173}$ .

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