Question

[Quách Trung Kiên]

Cho S=1/31+1/32+....+1/60 

        CM 3/5<S<4/5

Answer 1

[Lê Anh Tú]

• \(S=\frac{1}{31}+\frac{1}{32}+...+\frac{1}{60}\)

\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+..+\frac{1}{50}\right)+\left(\frac{1}{51}+..+\frac{1}{60}\right)\)

\(\Rightarrow S>\left(\frac{1}{40}+\frac{1}{40}+..+\frac{1}{40}\right)+\left(\frac{1}{50}+\frac{1}{50}+..+\frac{1}{50}\right)+\left(\frac{1}{60}+\frac{1}{60}+..+\frac{1}{60}\right)\)

\(\Rightarrow S>10\cdot\frac{1}{40}+10\cdot\frac{1}{50}+10\cdot\frac{1}{60}=\frac{37}{60}>\frac{36}{60}=\frac{3}{5}\left(1\right)\)

• \(S=\frac{1}{31}+\frac{1}{32}+..+\frac{1}{60}\)

\(S=\left(\frac{1}{31}+\frac{1}{32}+..+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+\frac{1}{52}+..+\frac{1}{60}\right)\)

\(S< \left(\frac{1}{31}+\frac{1}{31}+..+\frac{1}{31}\right)+\left(\frac{1}{41}+\frac{1}{41}+..+\frac{1}{41}\right)+\left(\frac{1}{51}+\frac{1}{51}+..+\frac{1}{51}\right)\)

\(S< 10\cdot\frac{1}{31}+10\cdot\frac{1}{41}+10\cdot\frac{1}{51}=\frac{10}{31}+\frac{10}{41}+\frac{10}{51}< \frac{10}{30}+\frac{10}{40}+\frac{10}{50}\)

\(S< \frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{47}{60}< \frac{48}{60}=\frac{4}{5}\left(2\right)\)

Từ (1) và (2) => đpcm