Tính tổng biểu thức: 1/3 + 1/3^2 + 1/3^3 + 1/3^4+ ... + 1/3^100
Giải thích
\[A = \frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + \frac{1}{{{3^4}}} + ... + \frac{1}{{{3^{100}}}}\]
\[3A = 1 + \frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + ... + \frac{1}{{{3^{99}}}}\]
\[3A - A = \left( {1 + \frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + ... + \frac{1}{{{3^{99}}}}} \right) - \left( {\frac{1}{3} + \frac{1}{{{3^2}}} + ... + \frac{1}{{{3^{99}}}} + \frac{1}{{{3^{100}}}}} \right)\]
\[2A = 1 - \frac{1}{{{3^{100}}}} = \frac{{{3^{100}} - 1}}{{{3^{100}}}}\]
Suy ra \[A = \frac{{{3^{100}} - 1}}{{2 \cdot {3^{100}}}}.\]