\(B=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)...\left(\frac{1}{400}-1\right)\)
\(-B=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)...\left(1-\frac{1}{400}\right)\)
\(-B=\frac{3}{4}\cdot\frac{5}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{399}{400}\)
\(-B=\frac{\left(1\cdot3\right)\left(2\cdot4\right)\left(3\cdot5\right)...\left(19\cdot21\right)}{\left(2\cdot2\right)\left(3\cdot3\right)\left(4\cdot4\right)...\left(20\cdot20\right)}\)
\(-B=\frac{\left(1\cdot2\cdot3\cdot...\cdot19\right)\left(3\cdot4\cdot5\cdot...\cdot21\right)}{\left(2\cdot3\cdot4\cdot...\cdot20\right)\left(2\cdot3\cdot4\cdot...\cdot20\right)}\)
\(-B=\frac{1\cdot21}{20\cdot2}\)
\(-B=\frac{21}{40}\)
\(B=\frac{-21}{40}\)
\(A=\frac{\text{2.3.5+2.2.3.3.5.5+2.5.3.7.2.2.2.5 }}{2.3.7+2.2.3.3.7.5+2.3.3.3.7.7+2.5.3.7.7.2.2.2}\)
\(=\frac{2.3.5.\left(1+2.3.5+2^3.5.7\right)}{2.3.7.\left(1+2.3.5+3.3.7+2^3.5.7\right)}\)
\(=\frac{2.3.5}{2.3.7.3.3.7}=\frac{5}{441}\)
