Question

[Phạm Nguyễn Tất Đạt]

Cho B=2/1*4/3*6/5*...*100/99.CMR 12<B<13

Answer 1

[soyeon_Tiểubàng giải]

Áp dụng \(\frac{a}{b}>1\Leftrightarrow\frac{a+m}{b+m}< \frac{a}{b}< \frac{a-m}{b-m}\) (a;b;m \(\in\) N*) ta có:

\(S=\frac{2}{1}.\frac{4}{3}.\frac{6}{5}.\frac{8}{7}.\frac{10}{9}...\frac{100}{99}\)

=> \(\frac{2}{1}.\frac{4}{3}.\frac{6}{5}.\frac{9}{8}.\frac{11}{10}....\frac{101}{100}< S< \frac{2}{1}.\frac{4}{3}.\frac{6}{5}.\frac{8}{7}.\frac{9}{8}...\frac{99}{98}\)

\(\Rightarrow\left(\frac{2}{1}.\frac{4}{3}.\frac{6}{5}\right)^2.\frac{8}{7}.\frac{9}{8}.\frac{10}{9}.\frac{11}{10}...\frac{100}{99}.\frac{101}{100}\) < S² \(< \left(\frac{2}{1}.\frac{4}{3}.\frac{6}{5}.\frac{8}{7}\right)^2.\frac{9}{8}.\frac{10}{9}...\frac{99}{98}.\frac{100}{99}\)

=> \(\left(\frac{16}{5}\right)^2.\frac{101}{7}\) < S² < \(\left(\frac{128}{35}\right)^2.\frac{100}{8}\)

=> 147 < S² < 167

=> 144 < S² < 169

=> 12² < S² < 13²

=> 12 < S < 13 (đpcm)

Answer 2

[Nguyễn Anh Duy]

* là dấu nhân à bạn??