\(A=\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot....\cdot\frac{960}{961}\)
\(=\frac{1.3}{2.2}\cdot\frac{2.4}{3.3}\cdot\frac{3.5}{4.4}\cdot.....\cdot\frac{30.32}{31.31}\)
\(=\frac{\left(1.2.3....30\right)\left(3.4.5....32\right)}{\left(2.3.4....31\right)\left(2.3.4...31\right)}\)
\(=\frac{32}{31.2}=\frac{32}{62}=\frac{16}{31}\)
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{960}{961}\)
\(\Rightarrow A=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{30.32}{31.31}\)
\(\Rightarrow A=\frac{1.2.3....30}{2.3.4....31}.\frac{3.4.5.....32}{2.3.4.....31}\)
\(\Rightarrow A=\frac{1}{31}.\frac{32}{2}=\frac{16}{31}\)
