Question

[Bùi Minh Đức]

Cho A= \(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{59}+\frac{1}{60}\)

chứng minh rằng: \(\frac{3}{5}\)<A < \(\frac{4}{5}\)

Answer 1

[Akai Haruma]

Lời giải:

$A=(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40})+(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50})+(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60})$

$> \frac{10}{40}+\frac{10}{50}+\frac{10}{60}=\frac{37}{60}> \frac{36}{60}=\frac{3}{5}(1)$

Lại có:

$A=(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40})+(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50})+(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60})$

$< \frac{10}{30}+\frac{10}{40}+\frac{10}{50}=\frac{47}{60}< \frac{48}{60}=\frac{4}{5}(2)$

Từ $(1); (2)\Rightarrow$ ta có đpcm.