Question

$\frac{x-2}{2} + \frac{x-2}{6} +...+ \frac{x-2}{2014ori2015} =2014$ rezolvare completa.

Answer 1

$\dfrac{x-2}{1\cdot2}+\dfrac{x-2}{2\cdot 3}+...+\dfrac{x-2}{2014\cdot 2015}= 2014 \\ \\ (x-2)\cdot \Big(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot 3}+...+\dfrac{1}{2014\cdot 2015}\Big)= 2014 \\ \\ (x-2)\cdot \Big(\dfrac{2-1}{1\cdot2}+\dfrac{3-2}{2\cdot 3}+...+\dfrac{2015-2014}{2014\cdot 2015}\Big) = 2014 \\ \\ (x-2) \cdot \\ \cdot \Big(\dfrac{2}{1\cdot 2}-\dfrac{1}{1\cdot 2}+\dfrac{3}{2\cdot 3}-\dfrac{2}{2\cdot 3}+...+\dfrac{2015}{2014\cdot 2015}-\dfrac{2014}{2014\cdot 2015}\Big)=2014$

$(x-2)\cdot \Big(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2014}-\dfrac{1}{2015}\Big) = 2014 \\ \\ (x-2)\cdot \Big( 1-\dfrac{1}{2015}\Big) = 2014 \\ \\ (x-2)\cdot \dfrac{2015-1}{2015} = 2014 \\ \\ (x-2)\cdot \dfrac{2014}{2015} = 2014\Big|:2014 \\ \\ (x-2)\cdot \dfrac{1}{2015} = 1 \\ \\ x-2 = 2015 \\ \\ x = 2015+2 \\ \\ x=2017$