\(A=\frac{5}{1x3}+\frac{5}{3x5}+...+\frac{5}{99x101}\)

\(A=\frac{5}{2}x\left(\frac{2}{1x3}+\frac{2}{3x5}+...+\frac{2}{99x101}\right)\)

\(A=\frac{5}{2}x\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}\right)\)

\(A=\frac{5}{2}x\left(1-\frac{1}{101}\right)\)

\(A=\frac{5}{2}x\frac{100}{101}\)

\(A=\frac{250}{101}\)