\(S=5+5^2+5^3+...+5^{100}\)
\(\Rightarrow5S=5^2+5^3+5^4+...+5^{101}\)
\(\Rightarrow5S-S=4S=\left(5^2+5^3+5^4+...+5^{101}\right)-\left(5+5^2+5^3+...+5^{100}\right)\)
\(4S=5^{101}-5\)
\(\Rightarrow S=\frac{5^{101}-5}{4}\)
\(a^1+a^2+a^3+...+a^n=\frac{a^{n+1}-a}{a-1}\)