\(S=1+5^2+5^4+...+5^{2014}\)
\(\Rightarrow25S=5^2+5^4+...+5^{2014}+5^{2016}\)
\(\Rightarrow25S-5^{2016}+1=1+5^2+5^4+...+5^{2014}\)
\(\Rightarrow25S-5^{2016}+1=S\)
\(\Rightarrow24S=5^{2016}-1\)
\(\Rightarrow S=\frac{1}{24}\left(5^{2016}-1\right)\)