Question

Rezolvă expresia 100 pct​

Answer 1

Explicație pas cu pas:

a)

$E_1(x)=\frac{x^3-x^2}{x^2-2x+1} =\frac{x^2(x-1)}{(x-1)^2} =\frac{x^2}{x-1} \ \forall x \in \mathbb{R}\setminus\{\pm1\}$

$E_1(-1)\cdot E_1(\frac{1}{2})=\frac{(-1)^2}{-1-1} \cdot \frac{(\frac{1}{2})^2 }{\frac{1}{2} -1} =\frac{1}{-2} \cdot \frac{\frac{1}{4} }{\frac{-1}{2} } =\frac{1}{-2} \cdot \frac{2}{-4} =\frac{1}{4}$

b)

$E_2(x)=\frac{(x-1)^2+(x+1)(x+2)+x-3}{x^2-1} =\frac{x^2-2x+1+x^2+3x+2+x-3}{x^2-1} =\frac{2x^2+2x}{x^2-1} =\frac{2x(x+1)}{(x-1)(x+1)} =\frac{2x}{x-1}$

$a=E_1(n)-\frac{1}{2} \cdot E_2(n)\\a=\frac{n^2}{n-1} -\frac{1}{2} \cdot \frac{2n}{n-1} =\frac{n^2-n}{n-1} =\frac{n(n-1)}{n-1} =n$

$\boxed{a=n \in \mathbb{N} \ \forall n \in \mathbb{N}, n\geq 2}$