Question

Chứng minh rằng: 1/20*23+1/23*26+1/26*29+....+1/77*80 <1/79

Answer 1

\(\frac{1}{20\cdot23}+\frac{1}{23\cdot26}+\frac{1}{26\cdot29}+...+\frac{1}{77\cdot80}\)

\(< \frac{1}{3}\left[\frac{3}{20\cdot23}+\frac{3}{23\cdot26}+\frac{3}{26\cdot29}+...+\frac{3}{77\cdot80}\right]\)

\(< \frac{1}{3}\left[\frac{1}{20}-\frac{1}{23}+\frac{1}{23}-\frac{1}{26}+...+\frac{1}{77}-\frac{1}{80}\right]\)

\(< \frac{1}{3}\left[\frac{1}{20}-\frac{1}{80}\right]\)

\(< \frac{1}{3}\left[\frac{4}{80}-\frac{1}{80}\right]\)

\(< \frac{1}{3}\cdot\frac{3}{80}=\frac{1}{80}< \frac{1}{79}(đpcm)\)