Rút gọn biểu thức: A = 1 + 1/2 + 1/2^2 + + 1/2^2012
Giải thích
\[A = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ... + \frac{1}{{{2^{2012}}}}\]
\[ \Leftrightarrow 2A = 2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ... + \frac{1}{{{2^{2011}}}}\]
\[ \Leftrightarrow 2A - A = \left( {2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ... + \frac{1}{{{2^{2011}}}}} \right) - \left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + ... + \frac{1}{{{2^{2012}}}}} \right)\]
\[ \Leftrightarrow 2A - A = 2 - \frac{1}{{{2^{2012}}}}\]
\[ \Leftrightarrow A = \frac{{{2^{2012}} + 1}}{{{2^{2012}}}}\]
Vậy \[A = \frac{{{2^{2012}} + 1}}{{{2^{2012}}}}\].