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\(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)
\(\Rightarrow A>\frac{1}{2}.\frac{2}{3}.\frac{4}{5}...\frac{98}{99}\)
\(\Rightarrow A^2>\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}...\frac{98}{99}.\frac{99}{100}\)
\(\Rightarrow A^2>\frac{1}{100}=\frac{1}{10^2}\)
Vậy \(A>\frac{1}{10}\)
\(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{9999}{10000}\)
\(\Rightarrow A>\frac{1}{2}.\frac{2}{3}.\frac{4}{5}...\frac{9998}{9999}\)
\(\Rightarrow A^2>\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}...\frac{9998}{9999}.\frac{9999}{10000}\)
\(\Rightarrow A^2>\frac{1}{10000}=\frac{1}{100^2}\)
\(VayA>\frac{1}{100}=B\)
\(P=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{399}{400}\)
\(\Rightarrow P< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{400}{401}\)
\(\Rightarrow P^2< \frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.\frac{5}{6}.\frac{6}{7}...\frac{399}{400}.\frac{400}{401}\)
\(\Rightarrow P^2< \frac{1}{401}< \frac{1}{400}=\frac{1}{20^2}\)
\(\Rightarrow P< \frac{1}{20}\)
P=1/2.3/4.5/6.....399/400
=>P<2/3.4/5......400/401
=>P2<1/2.2/3.3/4......398/399.399/400.400/401
=1/401<1/400=(1/20)2
=>P<1/20
\(D=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)
\(D< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{100}{101}\)
\(D^2< \frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.\frac{5}{6}.\frac{6}{7}...\frac{99}{100}.\frac{100}{101}\)
\(D^2< \frac{1}{101}< \frac{1}{100}=\left(\frac{1}{10}\right)^2\)
=> \(D< \frac{1}{10}\left(đpcm\right)\)
\(D=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)
\(D< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{100}{101}\)
\(D^2< \frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.\frac{5}{6}.\frac{6}{7}...\frac{99}{100}.\frac{100}{101}\)
\(D^2< \frac{1}{101}< \frac{1}{100}=\left(\frac{1}{10}\right)^2\)
\(= >D< \frac{1}{10}\)
\(\text{k tui}\)