\(B=\dfrac{1}{1\cdot6}+\dfrac{1}{2\cdot9}+\dfrac{1}{3\cdot12}+\dfrac{1}{4\cdot15}+\dfrac{1}{5\cdot18}+\dfrac{1}{6\cdot21}\)
\(=\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot3}+\dfrac{1}{3\cdot3\cdot4}+\dfrac{1}{4\cdot3\cdot5}+\dfrac{1}{5\cdot3\cdot6}+\dfrac{1}{6\cdot3\cdot7}\)
\(=\dfrac{1}{3}\cdot\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+\dfrac{1}{6\cdot7}\right)\)
\(=\dfrac{1}{3}\cdot\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\right)\)
\(=\dfrac{1}{3}\cdot\left(1-\dfrac{1}{7}\right)\)
\(=\dfrac{1}{3}\cdot\dfrac{6}{7}=\dfrac{2}{7}\)
