${A=}$ $\frac{25}{1.2.3}$ ${+...+}$ $\frac{25}{20.21.22}$
${=25.}$ $\frac{1}{2}$. ($\frac{1}{1.2}$-$\frac{1}{2.3}$+.....+$\frac{1}{20.21}$ - $\frac{1}{21.22}$)
${=}$ $\frac{25}{2}$. (${1-}$ $\frac{1}{21.22}$)
${=}$ $\frac{25}{2}$. $\frac{461}{462}$
${=}$ $\frac{11525}{924}$