\(A=1+\dfrac{1}{3}+\dfrac{1}{6}+...+\dfrac{1}{8\cdot\dfrac{7}{2}}\)
\(=1+2\left(\dfrac{1}{6}+\dfrac{1}{12}+...+\dfrac{1}{7\cdot8}\right)\)
\(=1+2\cdot\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{7}-\dfrac{1}{8}\right)\)
=1+2*3/8
=1+3/4=7/4