a=2/3+2/6+2/12+2/24+…+2/768

$\displaystyle  \begin{array}{{>{\displaystyle}l}}
A=\frac{2}{3} +\frac{2}{6} +\frac{2}{12} +\frac{2}{24} +...+\frac{2}{768}\\
2A=\frac{4}{3} +\frac{2}{3} +\frac{2}{6} +\frac{2}{12} +...+\frac{2}{384}\\
2A-A=\left(\frac{4}{3} +\frac{2}{3} +\frac{2}{6} +\frac{2}{12} +...+\frac{2}{384}\right) -\left(\frac{2}{3} +\frac{2}{6} +\frac{2}{12} +\frac{2}{24} +...+\frac{2}{768}\right)\\
A=\frac{4}{3} +\frac{2}{3} +\frac{2}{6} +\frac{2}{12} +...+\frac{2}{384} -\frac{2}{3} -\frac{2}{6} -\frac{2}{12} -\frac{2}{24} -...-\frac{2}{384} -\frac{2}{768}\\
A=\frac{4}{3} -\frac{2}{768}\\
A=\frac{511}{384}
\end{array}$