Question

A=1/5+1/5^2+1/5^3+.......+1/5^10

Answer 1

\(A=\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+...+\dfrac{1}{5^{10}}\)

\(5A=5\left(\dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+...+\dfrac{1}{5^{10}}\right)\)

\(5A=1+\dfrac{1}{5}+\dfrac{1}{5^2}+...+\dfrac{1}{5^9}\)

\(5A-A=\left(1+\dfrac{1}{5}+...+\dfrac{1}{5^9}\right)-\left(\dfrac{1}{5}+\dfrac{1}{5^2}+...+\dfrac{1}{5^{10}}\right)\)

\(4A=1-\dfrac{1}{5^{10}}\Rightarrow A=\dfrac{1-\dfrac{1}{5^{10}}}{4}\)