$\frac{5}{1.3}$+$\frac{5}{3.5}$+ $\frac{5}{5.7}$+......$\frac{5}{99.101}$
= 5. ($\frac{1}{1.3}$+$\frac{1}{3.5}$+$\frac{1}{99.101}$)
= $\frac{5}{2}$.($\frac{2}{1.3}$+$\frac{2}{3.5}$+$\frac{2}{99.101}$)
= $\frac{5}{2}$.(1-$\frac{1}{3}$+$\frac{1}{3}$- $\frac{1}{5}$+....+$\frac{1}{99}$-$\frac{1}{101}$)
= $\frac{5}{2}$.(1-$\frac{1}{101}$)
= $\frac{5}{2}$.$\frac{100}{101}$
= $\frac{250}{101}$