Question

Chứng minh

H = \(\frac{1}{2}< \frac{1}{51}+\frac{1}{52}+\frac{1}{53}+...+\frac{1}{100}< 1\\\)

Answer 1

Ta có :

\(H=\frac{1}{51}+\frac{1}{52}+\frac{1}{52}+....+\frac{1}{100}\)

\(\Rightarrow H>\frac{1}{100}+\frac{1}{100}+\frac{1}{100}+....+\frac{1}{100}\)

\(\Rightarrow H>\frac{1}{100}.50\)

\(\Rightarrow H>\frac{1}{2}\)

Lại có :

\(H=\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+.....+\frac{1}{100}\)

\(\Rightarrow H< \frac{1}{51}+\frac{1}{51}+\frac{1}{51}+........+\frac{1}{51}\)

\(\Rightarrow H< \frac{1}{51}.50\)

\(\Rightarrow H< \frac{50}{51}\)

\(\Rightarrow H< 1\)

Vậy \(\frac{1}{2}< H< 1\left(ĐPCM\right)\)