Question

Tính tổng cơ số A=3/5+3/5^4+3/5^7+...+3/5^100

giúp mik vs ạ mik đg cần gấp ạ

Answer 1

Ta có \(A=3\left(\dfrac{1}{5}+\dfrac{1}{5^4}+\dfrac{1}{5^7}+...+\dfrac{1}{5^{100}}\right)\Rightarrow5^3A=3\left(5^2+\dfrac{1}{5}+\dfrac{1}{5^4}+...+\dfrac{1}{5^{97}}\right)\Rightarrow5^3A-A=3\left[5^2+\left(\dfrac{1}{5}-\dfrac{1}{5}\right)+\left(\dfrac{1}{5^4}-\dfrac{1}{5^4}\right)+\left(\dfrac{1}{5^7}-\dfrac{1}{5^7}\right)+...+\left(\dfrac{1}{5^{97}}-\dfrac{1}{5^{97}}\right)-\dfrac{1}{5^{100}}\right]\Rightarrow624A=3\left(25-\dfrac{1}{5^{100}}\right)=3.\dfrac{5^{102}-1}{5^{100}}=\dfrac{3.5^{102}-3}{5^{100}}\Rightarrow A=\dfrac{3.5^{102}-3}{5^{100}.624}\)