Question

[Trịnh Thị Vân Anh]

4.5¹⁰⁰.(1/5+1/5²+1/5³+....+1/5¹⁰⁰)+1

Answer 1

[Mashiro Shiina]

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:

\(A=4.5^{100}.\dfrac{1-5^{100}}{4}+1\)

\(A=5^{100}.\left(1-5^{100}\right)+1\)

\(A=5^{100}-5^{200}+1\)