Question

Chứng minh rằng: S= 1/31+1/32+1/33+...+1/60<4/5

Answer 1

\(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}\)

\(\Rightarrow S=\left(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{45}\right)+\frac{1}{46}+\frac{1}{47}...+\frac{1}{60}\)

\(\Rightarrow S< \left(\frac{1}{30}+\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\right)+\frac{1}{46}+\frac{1}{47}...+\frac{1}{60}\)(15 số hạng \(\frac{1}{30}\))

\(\Rightarrow S< \frac{15}{30}+\frac{1}{46}+\frac{1}{47}...+\frac{1}{60}< \frac{1}{2}< \frac{4}{5}\)

Vậy \(S< \frac{4}{5}\)

Answer 2

S < 1/40 x 30 = 3/4 < 4/5 

 =) S < 4/5

  Vậy S < 4/5

 học tốt nha

Answer 3

\(\frac{1}{31}< \frac{1}{30};\frac{1}{32}< \frac{1}{30};\frac{1}{33}< \frac{1}{30};...;\frac{1}{40}< \frac{1}{30}\)

\(\Rightarrow\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{40}< \frac{1}{30}.10=\frac{1}{3}\left(1\right)\)

\(\frac{1}{41}< \frac{1}{40};\frac{1}{42}< \frac{1}{40};\frac{1}{43}< \frac{1}{40};...;\frac{1}{50}< \frac{1}{40}\)

\(\Rightarrow\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+...+\frac{1}{50}< \frac{1}{40}.10=\frac{1}{4}\left(2\right)\)

\(\frac{1}{51}< \frac{1}{50};\frac{1}{52}< \frac{1}{50};\frac{1}{53}< \frac{1}{50};...;\frac{1}{60}< \frac{1}{50}\)​

\(\Rightarrow\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+...+\frac{1}{60}< \frac{1}{50}.10=\frac{1}{5}\left(3\right)\)

Từ (1) ,(2) và (3) suy ra

\(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}< \frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{47}{60}< \frac{48}{60}=\frac{4}{5}\)

\(\Rightarrow S< \frac{4}{5}\)