\(\frac{5}{1\times3}+\frac{5}{3\times5}+\frac{5}{5\times7}+...+\frac{5}{97\times99}+\frac{5}{99\times101}\)
\(=\frac{5}{2}\times\left(\frac{2}{1\times3}+\frac{2}{3\times5}+\frac{2}{5\times7}+...+\frac{2}{97\times99}+\frac{2}{99\times101}\right)\)
\(=\frac{5}{2}\times\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{97}-\frac{1}{99}+\frac{1}{99}-\frac{1}{101}\right)\)
\(=\frac{5}{2}\times\left(1-\frac{1}{101}\right)\)
\(=\frac{5}{2}\times\frac{100}{101}\)
\(=\frac{250}{101}\)
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