\(B=\left(1+\frac{1}{1\times3}\right)\times\left(1+\frac{1}{2\times4}\right)\times\left(1+\frac{1}{3\times5}\right)\times...\times\left(1+\frac{1}{99\times101}\right)\)
\(B=\frac{1\times3+1}{1\times3}\times\frac{2\times4+1}{2\times4}\times\frac{3\times5+1}{3\times5}\times...\times\frac{99\times101+1}{99\times101}\)
\(B=\frac{2\times2}{1\times3}\times\frac{3\times3}{2\times4}\times\frac{4\times4}{3\times5}\times...\times\frac{100\times100}{99\times101}\)
\(B=\frac{\left(2\times3\times4\times...\times100\right)\times\left(2\times3\times4\times...\times100\right)}{\left(1\times2\times3\times...\times99\right)\times\left(3\times4\times5\times...\times101\right)}\)
\(B=\frac{100\times2}{101}=\frac{200}{101}\)
