\(A=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{100^2}=\frac{1}{2^2}.\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\)
\(>\frac{1}{2^2}.\left(1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4.5}+...+\frac{1}{50.51}\right)=\frac{1}{4}.\left(1+\frac{1}{4}+\frac{1}{9}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{50}-\frac{1}{51}\right)\)
\(=\frac{1}{4}.\left(1+\frac{1}{4}+\frac{1}{4}+\frac{1}{9}-\frac{1}{51}\right)>\frac{1}{4}.\left(1+\frac{1}{4}+\frac{1}{4}+\frac{1}{9}-\frac{1}{9}\right)=\frac{1}{4}.\left(1+\frac{1}{4}+\frac{1}{4}\right)=\frac{1}{4}.\frac{3}{2}=\frac{3}{8}\)
\(\Rightarrow A>\frac{3}{8}\left(đpcm\right)\)
