Chứng tỏ rằng: 1/2.3 + 1/3.4 +...+1/6.7 <1/2

* Hướng dẫn giải

$a)\frac{1}{2.3}+\frac{1}{3.4}+\mathrm{...}+\frac{1}{6.7}=\frac{1}{2}-\frac{1}{7}<\frac{1}{2}.$

$\begin{array}{l}b)\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+\frac{4}{13.17}+\frac{4}{17.21}=1-\frac{1}{21}<1.\\ \\ c)Tacó \frac{1}{2^2} <\frac{1}{1.2};\frac{1}{3^2}<\frac{1}{2.3};\frac{1}{4^2}<\frac{1}{3.4};\mathrm{...};\frac{1}{{10}^2}<\frac{1}{9.10}.\\ \\ Dođó,\frac{1}{2^2} +\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{{10}^2}<\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\mathrm{...}+\frac{1}{9.10}<1.\end{array}$