\[A = \frac{1}{5} + \frac{1}{{10}} + \frac{1}{{20}} + \frac{1}{{40}} + \frac{1}{{80}} + \frac{1}{{160}} + \frac{1}{{320}}\] $2 \times A = \frac{2}{5} + \frac{2}{{10}} + \frac{2}{{20}} + \frac{2}{{40}} + \frac{2}{{80}} + \frac{2}{{160}} + \frac{2}{{320}}$ $2 \times A = \frac{2}{5} + \frac{1}{5} + \frac{1}{{10}} + \frac{1}{{20}} + \frac{1}{{40}} + \frac{1}{{80}} + \frac{1}{{160}}$ $2 \times A - A = \frac{2}{5} - \frac{1}{{320}}$ $A = \frac{{127}}{{320}}$