Đặt:
\(A=4.5^{100}.\left(\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+.....+\dfrac{1}{5^{100}}\right)+1\)
\(S=\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+.....+\dfrac{1}{5^{100}}\)
\(5S=5\left(\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+.....+\dfrac{1}{5^{100}}\right)\)
\(5S=1+\dfrac{1}{5}+\dfrac{1}{5^2}+.....+\dfrac{1}{5^{99}}\)
\(5S-S=\left(1+\dfrac{1}{5}+\dfrac{1}{5^2}+.....+\dfrac{1}{5^{99}}\right)-\left(\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+....+\dfrac{1}{5^{100}}\right)\)\(4S=1-5^{100}\Rightarrow S=\dfrac{1-5^{100}}{4}\)
Thay S và A ta có:
\(A=4.5^{100}.\dfrac{1-5^{100}}{4}+1\)
\(A=5^{100}.\left(1-5^{100}\right)+1\)
\(A=5^{100}-5^{200}+1\)
