A=(200-2-1)(199-2-1)....(101-2-1)
\(A=\left(\frac{1}{200^2}-1\right)\left(\frac{1}{199^2}-1\right)\left(\frac{1}{198^2}-1\right).....\left(\frac{1}{101^2}-1\right)\)
\(A=\frac{1-200^2}{200^2}.\frac{1-199^2}{199^2}.\frac{1-198^2}{198^2}.....\frac{1-101^2}{101^2}\)
\(A=\frac{\left(1-200\right)\left(1+200\right)}{200^2}.\frac{\left(1-199\right)\left(1+199\right)}{199^2}.....\frac{\left(1-100\right)\left(1+100\right)}{100^2}.\frac{\left(1-101\right)\left(1+101\right)}{101^2}\)
\(A=\frac{-199.201}{200.200}.\frac{-198.200}{199.199}.\frac{-197.199}{198.198}.....\frac{-99.101}{100.100}.\frac{-100.102}{101.101}\)
\(A=\frac{199.201}{200.200}.\frac{198.200}{199.199}.\frac{197.199}{198.198}.....\frac{99.101}{100.100}.\frac{100.102}{101.101}\)
\(\Rightarrow A=\frac{200}{2.101}=\frac{201}{202}\)