Question

Ajutați-mă!!!!!!!!!!!​

Answer 1

$\Big[x+\dfrac{1}{2}\Big]+\Big[x-\dfrac{1}{2}\Big] = \big[2x\big]\\\\\\ \text{Identitatea lui Hermite:}\\ \\\big[x\big]+\Big[x+\dfrac{1}{2}\Big] = \big[2x\big]\\\\ \\ \Rightarrow \Big[x+\dfrac{1}{2}\Big]+\Big[x-\dfrac{1}{2}\Big] = \big[x\big]+\Big[x+\dfrac{1}{2}\Big]\\ \\ \Rightarrow \Big[x-\dfrac{1}{2}\Big] = \big[x\big]=k\in \mathbb{Z}\\ \\\\\big[u\big]\leq u<\big[x\big]+1\quad \text{(inegalitatea partii intregi)}$

$\\\Rightarrow k \leq x-\dfrac{1}{2}< k+1 \quad \text{si}\quad k \leq x < k + 1$

$\Rightarrow k+\dfrac{1}{2} \leq x < k+\dfrac{3}{2} \quad \text{si}\quad k \leq x < k + 1$

$\\\Rightarrow x\in \Big[k+\dfrac{1}{2},k+\dfrac{3}{2}\Big)\cap\Big[k,k+1\Big)\\ \\\\ \text{Dar, }\Big[k+\dfrac{1}{2},k+1\Big)\subset \Big[k,k+\dfrac{3}{2}\Big): \\ \\ \\\Rightarrow \boxed{x\in\Big[k+\dfrac{1}{2},k+1\Big),\quad k\in \mathbb{Z}\quad \text{(solutia finala)}}$