\(B=\dfrac{1}{6.10}+\dfrac{1}{10.14}+\dfrac{1}{14.18}+...+\dfrac{1}{402.406}\)
\(4B=\dfrac{4}{6.10}+\dfrac{4}{10.14}+\dfrac{4}{14.18}...+\dfrac{4}{402.406}\)
\(=\dfrac{1}{6}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{18}+...+\dfrac{1}{398}-\dfrac{1}{402}+\dfrac{1}{402}-\dfrac{1}{406}\)
\(=\dfrac{1}{6}-\dfrac{1}{406}=\dfrac{100}{609}\)
\(\Rightarrow B=\dfrac{25}{609}\)
\(B=\dfrac{1}{4}\cdot\left(\dfrac{4}{6\cdot10}+\dfrac{4}{10\cdot14}+...+\dfrac{4}{402\cdot406}\right)\)
=1/4(1/6-1/10+1/10-1/14+...+1/402-1/406)
=1/4*(1/6-1/406)
=1/4*400/(6*406)
=100/(6*406)=25/609
