## SUBJECT I

Solve the equation $\frac{x-2}{x}+\frac{x-4}{x}+\frac{x-6}{x}+\ldots . . .+\frac{2}{x}=12$
## SUBJECT I

Solve the equation $\frac{x-2}{x}+\frac{x-4}{x}+\frac{x-6}{x}+\ldots . . .+\frac{2}{x}=12$

(proposed by prof. Sîrb Vasile C.T. "A.P.I." Zalău)

Grading rubric

we set the condition $x \neq 0$.

The sum on the left side has terms in an arithmetic progression with the common difference $r=\frac{-2}{x}$

From the general term formula $a_{n}=a_{1}+(n-1) \cdot r$ we get $n=\frac{x-2}{2}$

From the sum $S_{n}=\frac{\left(a_{1}+a_{n}\right) \cdot n}{2}$ we have: $12=\frac{\left(\frac{x-2}{x}+\frac{2}{x}\right) \cdot \frac{x-2}{2}}{2}$ or $12=\frac{x-2}{4}$ $2 p$ from which $x=50$ $1 p$