Question

Cho A = \(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}\)

Hãy chứng minh A > 1

Answer 1

A=\(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}=\frac{1}{10}+\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}\right)\)

Ta thấy:

\(\frac{1}{10}>\frac{1}{100}\)

\(\frac{1}{11}>\frac{1}{100}\)

..................

\(\frac{1}{99}>\frac{1}{100}\)

\(\frac{1}{100}=\frac{1}{100}\)

Do đó, \(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}>\frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}\)

            \(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}>\frac{90}{100}\)

       \(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+...+\frac{1}{100}>\frac{1}{10}+\frac{90}{100}\)

                        \(A>\frac{100}{100}\)

                        A>1

Vậy A>1