Question

A= 1/4 + 1/4^2 + 1/4^3 +...+1/4^2013

CM: A < 1/3

Answer 1

Ta có :

\(A=\dfrac{1}{4}+\dfrac{1}{4^2}+\dfrac{1}{4^3}+.........+\dfrac{1}{4^{2013}}\)

\(\Leftrightarrow4A=1+\dfrac{1}{4}+\dfrac{1}{4^2}+.........+\dfrac{1}{4^{2012}}\)

\(\Leftrightarrow4A-A=\left(1+\dfrac{1}{4}+......+\dfrac{1}{4^{2012}}\right)-\left(\dfrac{1}{4}+\dfrac{1}{4^2}+........+\dfrac{1}{4^{2013}}\right)\)

\(\Leftrightarrow3A=1-\dfrac{1}{4^{2013}}\)

\(\Leftrightarrow A=\dfrac{1-\dfrac{1}{4^{2013}}}{3}\)