\(A=1+5+5^2+5^3+.....+5^{49}+5^{50}\)
\(\Rightarrow5A=5+5^2+5^3+5^4+...+5^{50}+5^{51}\)
\(5A-A=\left(5+5^2+5^3+5^4+...+5^{50}+5^{51}\right)-\left(1+5+5^2+5^3+....+5^{49}+5^{50}\right)\)
\(\Rightarrow4A=5^{51}-1\)
\(A=\frac{5^{51}-1}{4}\)
A = 1 + 5 + 52 + .... + 549 + 550
5A = 5 + 52 + 53 + ... + 550 + 551
5A - A = (5 + 52 + 53 + ... + 550 + 551) - (1 + 5 + 52 + .... + 549 + 550)
4A = 551 - 1
\(A=\frac{5^{51}-1}{4}\)