\(A=\left(1+5^2+5^4+...+5^{200}\right)\)
=>\(5^2.A=5^2.\left(1+5^2+5^4+...+5^{200}\right)\)
=>\(25A=5^2+5^4+5^8+...+5^{202}\)
=>\(25A-A=\left(5^2+5^4+5^6+...+5^{202}\right)-\left(1+5^2+5^4+...+5^{200}\right)\)
=>\(24A=5^{202}-1\Rightarrow A=\frac{5^{202}-1}{24}\)