Chứng minh rằng (frac{1}{{20.23}} + frac{1}{{23.26}} + frac{1}{{26.29}} + ... + frac{1}{{77.80}}...

* Hướng dẫn giải

+ Biến đổi      

\(\begin{array}{l}
\frac{1}{{20.23}} + \frac{1}{{23.26}} + \frac{1}{{26.29}} + ... + \frac{1}{{77.80}} =  = \frac{1}{3}.(\frac{3}{{20.23}} + \frac{3}{{23.26}} + \frac{3}{{26.29}} + ... + \frac{3}{{77.80}})\\
 = \frac{1}{3}.\left( {\frac{1}{{20}} - \frac{1}{{23}} + \frac{1}{{23}} - \frac{1}{{26}} + ... + \frac{1}{{77}} - \frac{1}{{80}}} \right)\\
 = \frac{1}{3}.\left( {\frac{1}{{20}} - \frac{1}{{80}}} \right)\\
 = \frac{1}{3}.\frac{3}{{80}} = \frac{1}{{80}}
\end{array}\)

Vì \(\frac{1}{{80}} < \frac{1}{9}\) nên  \(\frac{1}{{20.23}} + \frac{1}{{23.26}} + \frac{1}{{26.29}} + ... + \frac{1}{{77.80}} < \frac{1}{9}\)