Question

rút gọn biểu thức sau:

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

 

Answer 1

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

\(2A=2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}\)

\(2A-A=\left(2+1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2011}}\right)-\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)\)

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

Answer 2

Dat B=1/2+1/2^2+1/2^3+.............+1/2^2012=>2B=1+1/2+1/2^2+.......+1/2^2011

=>2B-B=1/2^2011-1/2^2012=2^2012-2^2011/2^4023

=>A=1+2^2012-2^2011/2^4023

Answer 3

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

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

\(\Rightarrow2A-A=\left(2+1+\frac{1}{2}+...+\frac{1}{2^{2011}}\right)-\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+..+\frac{1}{2^{2012}}\right)\)

\(\Rightarrow A=2-\frac{1}{2^{2012}}\)

\(\Rightarrow A=\frac{2^{2013}}{2^{2012}}-\frac{1}{2^{2012}}=\frac{2^{2013}-1}{2^{2012}}\)

Vậy \(A=\frac{2^{2013}-1}{2^{2012}}\)