Chứng minh biểu thức A = 1/10 + 1/11 + 1/12 + . . . + 1/99 + 1/100 > 1.

Hướng dẫn giải:

Ta có: \(\frac{1}{{10}} > \frac{1}{{20}};\)

\(\frac{1}{{11}} > \frac{1}{{20}};\)

\(\frac{1}{{12}} > \frac{1}{{20}};\)

….

\(\frac{1}{{20}} = \frac{1}{{20}}.\)

Cộng vế theo vế ta được:

\(\frac{1}{{10}} + \frac{1}{{11}} + \frac{1}{{12}} + ... + \frac{1}{{20}} > \frac{1}{{20}} + \frac{1}{{20}} + ... + \frac{1}{{20}} = 10 \cdot \frac{1}{{20}} = \frac{1}{2};\)

Tương tự ta có:

\(\frac{1}{{21}} + \frac{1}{{22}} + \frac{1}{{23}} + ... + \frac{1}{{30}} > \frac{1}{{30}} + \frac{1}{{30}} + ... + \frac{1}{{30}} = 10 \cdot \frac{1}{{30}} = \frac{1}{3};\)

\(\frac{1}{{31}} + \frac{1}{{32}} + \frac{1}{{33}} + ... + \frac{1}{{40}} > \frac{1}{{40}} + \frac{1}{{40}} + ... + \frac{1}{{40}} = 10 \cdot \frac{1}{{40}} = \frac{1}{4}.\)

Suy ra \(\frac{1}{{10}} + \frac{1}{{11}} + \frac{1}{{12}} + ... + \frac{1}{{39}} + \frac{1}{{40}} > \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{{6 + 4 + 3}}{{12}} = \frac{{13}}{{12}} > 1.\)

Do đó \(A = \frac{1}{{10}} + \frac{1}{{11}} + \frac{1}{{12}} + ... + \frac{1}{{99}} + \frac{1}{{100}} > 1.\)

Vậy \(A = \frac{1}{{10}} + \frac{1}{{11}} + \frac{1}{{12}} + ... + \frac{1}{{99}} + \frac{1}{{100}} > 1.\)