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\(\Leftrightarrow2\left(\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+...+\dfrac{1}{x\left(x+1\right)}\right)=\dfrac{2010}{2012}\)
\(\Leftrightarrow\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{2010}{4024}=\dfrac{1005}{2012}\)
=>1/x+1=-251/1006
=>x+1=-1006/251
=>x=-1257/251
\(\frac{1}{10}+\frac{1}{15}+\frac{1}{21}+...+\frac{2}{x\left(x+1\right)}=\frac{2010}{2012}\)
\(\frac{2}{20}+\frac{2}{30}+\frac{2}{42}+...+\frac{2}{x\left(x+1\right)}=\frac{2010}{2012}\)
\(\frac{2}{4.5}+\frac{2}{5.6}+\frac{2}{6.7}+...+\frac{2}{x\left(x+1\right)}=\frac{2010}{2012}\)
\(2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2010}{2012}\)
\(2\left(\frac{1}{4}-\frac{1}{x+1}\right)=\frac{2010}{2012}\)
\(\frac{1}{4}-\frac{1}{x+1}=\frac{2010}{2012}\div2\)
\(\frac{1}{4}-\frac{1}{x+1}=\frac{1005}{2012}\)
\(\frac{1}{x+1}=\frac{1}{4}-\frac{1005}{2012}\)
\(\frac{1}{x+1}=\frac{-502}{2012}=-\frac{251}{1006}\)
\(\Rightarrow x+1=1\div-\frac{251}{1006}=-\frac{1006}{251}\)
\(x=\frac{-1006}{251}-1=-\frac{1257}{251}\)
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