Question

1/3+1/6+1/10+...+2/x.(x+1)=2016/2018 

Answer 1

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2016}{2018}\)

\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}=\frac{504}{1009}\)

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}=\frac{504}{1009}\)

\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{504}{1009}\)

\(\frac{1}{2}-\frac{1}{x+1}=\frac{504}{1009}\)

\(\frac{1}{x+1}=\frac{1}{2}-\frac{504}{1009}\)

\(\frac{1}{x+1}=\frac{1}{2018}\)

\(\Rightarrow x+1=2018\)

\(\Rightarrow x=2017\)