Question

Rezolvati Ecuatia :

x+3/2 + x+4/3 + x+5/4 + x+5/4 + x+6/5 + x+7/6 = 5

Answer 1

[tex]\dfrac{x+3}{2}+\dfrac{x+4}{3}+\ldots+\dfrac{x+7}{6}=5\\ \dfrac{x+3}{2}+\dfrac{x+4}{3}+\ldots +\dfrac{x+7}{6}-5=0\\ \left(\dfrac{x+3}{2}-1\right)+\left(\dfrac{x+4}{3}-1\right)+\ldots+\left(\dfrac{x+7}{6}-1\right)=0\\ \dfrac{x+3-2}{2}+\dfrac{x+4-3}{3}+\ldots+\dfrac{x+7-6}{6}=0\\ \dfrac{x+1}{2}+\dfrac{x+1}{3}+\ldots+\dfrac{x+1}{6}=0\\ (x+1)\underbrace{\left(\dfrac{1}{2}+\dfrac{1}{3}+\ldots+\dfrac{1}{7}\right)}=0\\ ~~~~~~~~~~~~~~~~~~~~~~~~\neq 0\\ \text{Prin urmare:}\\ x+1=0\Rightarrow \boxed{x=-1} [/tex]