Question

rezolvați în mulțimea numerelor raționale ecuatia:
2x+1/2+2x+2/3+2x+3/4+2x+4/5+....+2x+(n-1)/n+2x+n/n+1=n

unde n apartine N*,n fixat.
VA ROG FRUMOS AJUTAȚI-MAAA​

Answer 1

$\it \dfrac{2x+1}{2}+\dfrac{2x+2}{3}+\dfrac{2x+3}{4}+\ ...\ +\dfrac{2x+n}{n+1}=n \Rightarrow\\ \\ \\ \Rightarrow \dfrac{2x+1}{2}+\dfrac{2x+2}{3}+\dfrac{2x+3}{4}+\ ...\ +\dfrac{2x+n}{n+1}=\underbrace{\it1+1+1+\ ...\ +1}_{n\ termeni} \Rightarrow\\ \\ \\ \Rightarrow\bigg(\dfrac{2x+1}{2}-1\bigg)+\bigg(\dfrac{2x+2}{3}-1\bigg)+\bigg(\dfrac{2x+3}{4}-1\bigg) +...+\bigg(\dfrac{2x+n}{n+1}-1\bigg)=0\Rightarrow$

$\it \Rightarrow \bigg(\dfrac{2x-1}{2}\bigg)+\bigg(\dfrac{2x-1}{3}\bigg)+\bigg(\dfrac{2x-1}{4}\bigg) +...+\bigg(\dfrac{2x-1}{n+1}\bigg)=0 \Rightarrow\\ \\ \\ \Rightarrow (2x-1)\bigg(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\ ...\ \dfrac{1}{n+1}\bigg)=0 \Rightarrow 2x-1=0 \Rightarrow 2x=1 \Rightarrow x=\dfrac{1}{2}$