\(A\cdot\left(1-\frac{1}{2}\right)=\left(1-\frac{1}{2}\right)\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}\right)=1-\frac{1}{2^{2013}}\)
\(\Rightarrow A=2\cdot\left(1-\frac{1}{2^{2013}}\right)=2-\frac{1}{2^{2012}}\)
Cũng có cách khác như sau:
\(A+\frac{1}{2^{2012}}=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2012}}+\frac{1}{2^{2012}}\)
\(A+\frac{1}{2^{2012}}=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2011}}+\frac{1}{2^{2011}}\)
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\(A+\frac{1}{2^{2012}}=2\)=> \(A=2-\frac{1}{2^{2012}}\)
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