\(5+5^3+5^5+...+5^{97}+5^{99}\)
\(\Rightarrow25A=5^2\left(5+5^3+5^5+...+5^{99}\right)\)
\(\Rightarrow25A=5^3+5^5+5^7+...+5^{101}\)
\(\Rightarrow25A-A=5^3+5^5+...+5^{99}+5^{101}-5-5^3-5^5-...-5^{99}\)
\(\Rightarrow24A=5^{101}-5\)
\(\Rightarrow A=\dfrac{5^{101}-5}{24}\)
