\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)...\left(\frac{1}{100^2-1}\right)\)
\(-A=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)...\left(1-\frac{1}{100^2}\right)\)
\(-A=\frac{3}{4}\cdot\frac{8}{9}\cdot...\cdot\frac{9999}{10000}\)
\(-A=\frac{\left(1\cdot3\right)\left(2\cdot4\right)...\left(99\cdot101\right)}{\left(2\cdot2\right)\left(3\cdot3\right)...\left(100\cdot100\right)}\)
\(-A=\frac{\left(1\cdot2\cdot...\cdot99\right)\left(3\cdot4\cdot...101\right)}{\left(2\cdot3\cdot...\cdot100\right)\left(2\cdot3\cdot...\cdot100\right)}\)
\(-A=\frac{1\cdot101}{100\cdot2}\)
\(-A=\frac{101}{200}\)
\(A=\frac{-101}{200}\)
