\(\left(x-\dfrac{1}{3}\right)^2-\dfrac{1}{4}=0\)

\(\Rightarrow\)\(\left(x-\dfrac{1}{3}\right)^2=0+\dfrac{1}{4}\)

\(\Rightarrow\)\(\left(x-\dfrac{1}{3}\right)^2=\dfrac{1}{4}\)

\(x-\dfrac{1}{3}=\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{1}{2}\)

\(x=\dfrac{1}{2}-\dfrac{1}{3}\)

\(\Rightarrow x=\dfrac{5}{6}\)

\(\dfrac{1}{10}+\dfrac{1}{40}+\dfrac{1}{88}+\dfrac{1}{154}+\dfrac{1}{238}+\dfrac{1}{340}\)

\(=\left(\dfrac{1}{2.5}+\dfrac{1}{5.8}+....+\dfrac{1}{14.17}+\dfrac{1}{17.20}\right)\div3\)

\(=\left(\dfrac{1}{2}-\dfrac{1}{20}\right)\div3\)

\(=\dfrac{9}{20}\div3\)

= \(\dfrac{3}{20}\)