Question

Cho A=\(\frac{50}{111}+\frac{50}{112}+\frac{50}{113}+\frac{50}{114}\). Chứng tơ \(1< A< 2\)

Answer 1

Ta có :

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

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

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

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

\(\Rightarrow A>\frac{50}{200}+\frac{50}{200}+\frac{50}{200}+\frac{50}{200}\)hay \(A>\frac{50}{200}.4\left(1\right)\)

Mặt khác :

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

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

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

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

\(\Rightarrow A< \frac{50}{100}+\frac{50}{100}+\frac{50}{100}+\frac{50}{100}\)hay \(A< \frac{50}{100}.4\left(2\right)\)

Từ \(\left(1\right)\)và \(\left(2\right)\Rightarrow1< A< 2\left(đpcm\right)\)