Question

Cho A=50/111+50/112+50/113+50/114.Chứng tỏ 1<A<2

Answer 1

Ta có :

\(A=\dfrac{50}{111}+\dfrac{50}{112}+\dfrac{50}{113}+\dfrac{50}{114}\)

Ta thấy :

\(\dfrac{50}{111}>\dfrac{50}{200}\)

\(\dfrac{50}{112}>\dfrac{50}{200}\)

\(\dfrac{50}{113}>\dfrac{50}{200}\)

\(\dfrac{50}{114}>\dfrac{50}{200}\)

\(\Rightarrow A>\dfrac{50}{200}+\dfrac{50}{200}+\dfrac{50}{200}+\dfrac{50}{200}\)

\(\Rightarrow A>\dfrac{50}{200}.4=1\) \(\left(1\right)\)

Mặt khác :

\(\dfrac{50}{111}< \dfrac{50}{100}\)

\(\dfrac{50}{112}< \dfrac{50}{100}\)

\(\dfrac{50}{113}< \dfrac{50}{100}\)

\(\dfrac{50}{114}< \dfrac{50}{100}\)

\(\Rightarrow A< \dfrac{50}{100}+\dfrac{50}{100}+\dfrac{50}{100}+\dfrac{50}{100}\)

\(\Rightarrow A< \dfrac{50}{100}.4=2\) \(\left(2\right)\)

Từ \(\left(1\right)+\left(2\right)\Rightarrow1< A< 2\rightarrowđpcm\)