Bài giải :

Ta biến đổi mẫu số :

${{25}^{30}}+{{25}^{28}}+{{25}^{26}}+...+{{25}^{2}}+1$                                                  

$=\left( {{25}^{28}}+{{25}^{24}}+{{25}^{20}}+...+{{25}^{4}}+1 \right)+\left( {{25}^{30}}+{{25}^{26}}+{{25}^{22}}+...+{{25}^{2}} \right)$

$=\left( {{25}^{28}}+{{25}^{24}}+{{25}^{20}}+...+{{25}^{4}}+1 \right)+{{25}^{2}}\left( {{25}^{28}}+{{25}^{24}}+{{25}^{20}}+...+1 \right)$

$=\left( {{25}^{28}}+{{25}^{24}}+{{25}^{20}}+...+{{25}^{4}}+1 \right)\left( 1+{{25}^{2}} \right)$

Suy ra $A=\frac{{{25}^{28}}+{{25}^{24}}+{{25}^{20}}+...+{{25}^{4}}+1}{\left( {{25}^{28}}+{{25}^{24}}+{{25}^{20}}+...+{{25}^{4}}+1 \right)\left( 1+{{25}^{2}} \right)}$

Vậy $A=\frac{1}{1+{{25}^{2}}}=\frac{1}{626}$