A=1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8
=1/2-1/8=3/8
\(=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{6\cdot7}+\frac{1}{7\cdot8}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}\)\(+\frac{1}{7}-\frac{1}{8}\)
=\(\left(\frac{1}{2}-\frac{1}{8}\right)+\left(\frac{1}{3}-\frac{1}{3}\right)+...+\left(\frac{1}{7}-\frac{1}{7}\right)\)
\(=\left(\frac{1}{2}-\frac{1}{8}\right)+0+...+0\)
\(=\frac{1}{2}-\frac{1}{8}=\frac{4}{8}-\frac{1}{8}=\frac{3}{8}\)
