Rút gọn A = 1 + 1/2 + (1/2)^2 + (1/2)^3 + + (1/2)^2012
Rút gọn A = 1 + \(\frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} + {\left( {\frac{1}{2}} \right)^3} + .... + {\left( {\frac{1}{2}} \right)^{2012}}\).
A = 1 + \(\frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} + {\left( {\frac{1}{2}} \right)^3} + .... + {\left( {\frac{1}{2}} \right)^{2012}}\)
2A = \[2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^{2011}}}}\]
2A – A = \[\left( {2 + 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^{2011}}}}} \right) - \left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ... + \frac{1}{{{2^{2012}}}}} \right)\]
A = \(2 - \frac{1}{{{2^{2012}}}} = \,\frac{{{2^{2013}} - 1}}{{{2^{2012}}}}\, = \,\frac{{{2^{2012}} + 1}}{{{2^{2012}}}}\).