\(A=\dfrac{2}{3\cdot5}+\dfrac{11}{5\cdot6}-\dfrac{3}{6\cdot9}-\dfrac{21}{9\cdot12}+\dfrac{3}{12\cdot15}\)
\(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}+\dfrac{1}{6}-\dfrac{1}{6}+\dfrac{1}{9}-\dfrac{1}{9}-\dfrac{1}{12}+\dfrac{1}{12}-\dfrac{1}{15}\)
\(=\dfrac{1}{3}-\dfrac{1}{15}=\dfrac{4}{15}\)