\(A=\frac{3}{2^2}\cdot\frac{8}{3^2}\cdot\frac{15}{4^2}\cdot...\cdot\frac{899}{30^2}\)

\(=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{899}{900}\)

\(=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot\frac{3\cdot5}{4\cdot4}\cdot...\cdot\frac{29\cdot31}{30\cdot30}\)

\(=\frac{1\cdot2\cdot3\cdot...\cdot29}{2\cdot3\cdot4\cdot...\cdot30}\cdot\frac{3\cdot4\cdot5\cdot...\cdot31}{2\cdot3\cdot4\cdot...\cdot30}=\frac{1}{30}\cdot\frac{31}{2}=\frac{31}{60}\)

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\(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.........\frac{899}{30^2}\)

A= \(\frac{1.3.2.4.3.5...29.31}{2.2.3.3...30.30}\)

A=\(\frac{\left(2.3...29.30\right)\left(3.4.5...29.31\right)}{\left(2.3.4...30\right)\left(2.3.4...30\right)}\)

A=\(\frac{31}{2.30}\)

A=\(\frac{31}{60}\)

\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}\frac{3.5}{4.4}.....\frac{29.31}{30.30}=\frac{\left(1.2.3....29\right)\left(3.4.5....31\right)}{\left(2.3.4....30\right)\left(2.3.4....30\right)}=\frac{31}{30.2}=\frac{31}{60}\)

\(A=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}...\frac{899}{30^2}=\frac{1.3.2.4.3.5...29.31}{2.2.3.3.4.4...30.30}=\frac{\left(2.3...29.30\right).\left(3.4.5...29.31\right)}{\left(2.3...29.30\right).\left(2.3.4...29.30\right)}=\frac{31}{2.30}=\frac{31}{60}\)

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