Chọn giá trị đúng của \(F = \frac{1}{2} + \frac{2}{{{2^2}}} + \frac{3}{{{2^3}}} + \frac{4}{{{2^4}}} + \frac{5}{{{2^5}}} + \ldots + \frac{{100}}{{{2^{100}}}} \)
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Lời giải:
Báo saiTa có
\(\begin{array}{l} 2F = 1 + \frac{2}{2} + \frac{3}{{{2^2}}} + \frac{4}{{{2^3}}} + \frac{5}{{{2^4}}} + \ldots + \frac{{99}}{{{2^{100}}}} + \frac{{100}}{{{2^{99}}}}\\ 2F - F = 1 + \left( {\frac{2}{2} - \frac{1}{2}} \right) + \left( {\frac{3}{{{2^2}}} - \frac{2}{{{2^2}}}} \right) + \left( {\frac{4}{{{2^3}}} - \frac{3}{{{2^3}}}} \right) + \ldots + \left( {\frac{{100}}{{{2^{99}}}} - \frac{{99}}{{{2^{99}}}}} \right) - \frac{{100}}{{{2^{100}}}}\\ F = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{99}}}} - \frac{{100}}{{{2^{100}}}} \end{array}\)
\(\begin{array}{l} Đặt\,A = 1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + \ldots + \frac{1}{{{2^{99}}}} = 2 - \frac{1}{{{2^{99}}}}\\ \Rightarrow F = 2 - \frac{1}{{{2^{99}}}} - \frac{{100}}{{{2^{100}}}} = \frac{{{2^{101}} - 102}}{{{2^{100}}}} \end{array}\)
