\(A=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot...\cdot\frac{29\cdot31}{30\cdot30}=\frac{1\cdot2\cdot...\cdot29}{2\cdot3\cdot...\cdot30}\cdot\frac{3\cdot4\cdot...\cdot31}{2\cdot3\cdot...\cdot30}=\frac{1}{30}\cdot\frac{31}{2}\)
\(=\frac{31}{60}\)
\(A=\frac{3}{2^2}\cdot\frac{8}{3^2}\cdot\frac{15}{4^2}\cdot...\frac{899}{30^2}\)
\(=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot\frac{3\cdot5}{4\cdot4}\cdot...\cdot\frac{29\cdot31}{30\cdot30}\)
\(=\frac{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot29\cdot31}{2\cdot2\cdot3\cdot3\cdot4\cdot4\cdot...\cdot30\cdot30}\)
\(=\frac{1\cdot2\cdot3\cdot...\cdot29}{2\cdot4\cdot6\cdot...\cdot30}\cdot\frac{3\cdot4\cdot5\cdot...\cdot31}{2\cdot3\cdot4\cdot...\cdot30}\)
\(=\frac{1}{30}\cdot\frac{31}{2}=\frac{31}{60}\)
