\(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{2014}{2016}\)
\(\Leftrightarrow\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{20}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{2014}{2016}\)
\(\Leftrightarrow\dfrac{2}{2\cdot3}+\dfrac{2}{3\cdot4}+\dfrac{2}{4\cdot5}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{2014}{2016}\)
\(\Leftrightarrow\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{1007}{2016}\)
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{1007}{2016}\)
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{1007}{2016}\)\(\Leftrightarrow\dfrac{1}{x+1}=\dfrac{1}{2016}\)
\(\Leftrightarrow x+1=2016\Leftrightarrow x=2015\)
