Câu hỏi của Mai Hiệp Đức

Question

Cho M= 1/2*3/4*5/6 * ... * 99/100

Chứng minh M< 1/10

I don't need you · Accepted Answer

Đặt $N=\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{100}{101}$

ta có: $M.N=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.\frac{5}{6}.\frac{6}{7}...\frac{99}{100}.\frac{100}{101}=\frac{1}{101}$

ta có: $\frac{1}{2}< \frac{2}{3};\frac{3}{4}< \frac{4}{5};\frac{5}{6}< \frac{6}{7};...;\frac{99}{100}< \frac{100}{101}$

$\Rightarrow M=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}< N=\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{100}{101}$

$\Rightarrow M.M< M.N$

$\Rightarrow M^2< \frac{1}{101}< \frac{1}{100}=\left(\frac{1}{10}\right)^2$

$\Leftrightarrow M^2< \left(\frac{1}{10}\right)^2$

$\Rightarrow M< \frac{1}{10}\left(đpcm\right)$

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