Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
S = 1 + \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + .... + {\left( {\frac{2}{3}} \right)^n}\\
\Rightarrow \frac{2}{3}S = \frac{2}{3} + {\left( {\frac{2}{3}} \right)^2} + {\left( {\frac{2}{3}} \right)^3} + .... + {\left( {\frac{2}{3}} \right)^{n + 1}}\\
\Rightarrow \frac{2}{3}S - S = \left[ {\frac{2}{3} + {{\left( {\frac{2}{3}} \right)}^2} + {{\left( {\frac{2}{3}} \right)}^3} + .... + {{\left( {\frac{2}{3}} \right)}^{n + 1}}} \right] - \left[ {1 + \frac{2}{3} + {{\left( {\frac{2}{3}} \right)}^2} + .... + {{\left( {\frac{2}{3}} \right)}^n}} \right]\\
\Leftrightarrow \frac{{ - 1}}{3}S = {\left( {\frac{2}{3}} \right)^{n + 1}} - 1\\
\Rightarrow S = 3 - 3.{\left( {\frac{2}{3}} \right)^{n + 1}}
\end{array}\)
