Chứng minh rằng 1/1.2 + 1/2.3 + 1/{3.4 + ... + \1/99.100 < 1
Giải thích
Ta có \(\frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + ... + \frac{1}{{99.100}}\)
\( = \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{{99}} - \frac{1}{{100}}\)
\( = 1 + \left( {\frac{1}{2} - \frac{1}{2}} \right) + \left( {\frac{1}{3} - \frac{1}{3}} \right) + ... + \left( {\frac{1}{{99}} - \frac{1}{{99}}} \right) - \frac{1}{{100}}\)
\( = 1 - \frac{1}{{100}}\)\( = \frac{{99}}{{100}} < 1\).
Vậy \(\frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + ... +
\frac{1}{{99.100}} < 1\).