Question

Thu gọn:D= \(\dfrac{1}{5}\)-\(\dfrac{1}{5^2}\)+\(\dfrac{1}{5^3}\)-\(\dfrac{1}{5^4}\)+\(\dfrac{1}{5^5}\)-...- \(\dfrac{1}{5^{100}}\)+\(\dfrac{1}{5^{101}}\)

Answer 1

Ta có :

\(D=\dfrac{1}{5}-\dfrac{1}{5^2}+\dfrac{1}{5^3}-\dfrac{1}{5^4}+\dfrac{1}{5^5}-..........-\dfrac{1}{5^{100}}+\dfrac{1}{5^{101}}\)

\(5D=1-\dfrac{1}{5}+\dfrac{1}{5^2}-\dfrac{1}{5^3}+\dfrac{1}{5^4}-\dfrac{1}{5^5}+..........+\dfrac{1}{5^{100}}\)

\(5D+D=\left(1-\dfrac{1}{5}+\dfrac{1}{5^2}-\dfrac{1}{5^3}+.........+\dfrac{1}{5^{100}}\right)+\left(\dfrac{1}{5}-\dfrac{1}{5^2}+..............-\dfrac{1}{5^{100}}+\dfrac{1}{5^{101}}\right)\)\(6D=1-\dfrac{1}{5^{101}}\)

\(D=\dfrac{1-\dfrac{1}{5^{101}}}{6}\)