tính tổng (1 5 5^1 ... 5^100)/(1 4 4^1 .. 4^100

\(S = \frac{1}{5} + \frac{1}{{{5^2}}} + \frac{1}{{{5^3}}} + ... + \frac{1}{{{5^{100}}}}\)

\(5S = 1 + \frac{1}{5} + \frac{1}{{{5^2}}} + ... + \frac{1}{{{5^{99}}}}\)

\(5S - S = \left( {1 + \frac{1}{5} + \frac{1}{{{5^2}}} + ... + \frac{1}{{{5^{99}}}}} \right) - \left( {\frac{1}{5} + \frac{1}{{{5^2}}} + \frac{1}{{{5^3}}} + ... + \frac{1}{{{5^{100}}}}} \right)\)

\(4S = 1 - \frac{1}{{{5^{100}}}}\)

\(4S = \frac{{{5^{100}} - 1}}{{{5^{100}}}}\)

\(S = \frac{{{5^{100}} - 1}}{{{{4.5}^{100}}}}\)