Question

Rút gọn biểu thức:

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

Answer 1

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

\(\Rightarrow2A=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2011}}\)

\(\Leftrightarrow2A-A=\left(2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2011}}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2012}}\right)\)

\(\Leftrightarrow A=2-\dfrac{1}{2^{2012}}\)

Answer 2

Ta có :

\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+....................+\dfrac{1}{2^{2012}}\)

\(\Leftrightarrow2A=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+................+\dfrac{1}{2^{2011}}\)

\(\Leftrightarrow2A-A=\left(2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+..........+\dfrac{1}{2^{2011}}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+............+\dfrac{1}{2^{2012}}\right)\)\(\Leftrightarrow A=2-\dfrac{1}{2^{2012}}\)