\(P=1+5^3+5^6+5^9+..................+5^{99}\)
\(\Leftrightarrow5^3P=125P=5^3+5^6+................+5^{99}+5^{102}\)
\(\Leftrightarrow125P-P=\left(5^3+5^6+...........+5^{102}\right)-\left(1+5^3+.........+5^{99}\right)\)
\(\Leftrightarrow124P=5^{102}-1\)
\(\Leftrightarrow P=\dfrac{5^{102}-1}{24}\)