Question

Cho biểu thức:

A = \(\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+\frac{1}{24}+...+\frac{1}{40}\)

Hãy chứng tỏ \(\frac{1}{2}\) < A < 1 

Answer 1

Ta có:

\(A=\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+...+\frac{1}{40}>\frac{1}{40}+\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}=\frac{20}{40}=\frac{1}{2}\)

=>A>\(\frac{1}{2}\)  (*)

Ta có:

\(A=\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+...+\frac{1}{40}< \frac{1}{20}+\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}=\frac{20}{20}=1\)

=>A<1  (**)

Từ (*) và (**) => \(\frac{1}{2}< A< 1\)

Answer 2

\(A=\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+\frac{1}{24}+...+\frac{1}{40}>\frac{1}{40}+\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\)

                                                                                         20 phân số 1/40

\(A>20x\frac{1}{40}=\frac{1}{2}\)

\(A=\frac{1}{21}+\frac{1}{22}+\frac{1}{23}+\frac{1}{24}+...+\frac{1}{40}< \frac{1}{20}+\frac{1}{20}+\frac{1}{20}+...+\frac{1}{20}\)

                                                                                           20 phân số 1/20

\(A< 20x\frac{1}{20}=1\)

Chứng tỏ 1/2 < A < 1