Chứng minh rằng: A = 1/12 + 1/13 + 1/14 + ... + 1/22 > 1/2

* Hướng dẫn giải

$A=\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\mathrm{...}+\frac{1}{22}>\frac{1}{2}$$\iff \frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\mathrm{...}+\frac{1}{22}>\frac{11}{22}$$\iff \left(\frac{1}{12}-\frac{1}{22}\right)+\left(\frac{1}{13}-\frac{1}{22}\right)+\left(\frac{1}{14}-\frac{1}{22}\right)+\mathrm{...}+\left(\frac{1}{22}-\frac{1}{22}\right)>0$

Vì $\frac{1}{12}>0,\frac{1}{13}>0,\mathrm{...},\frac{1}{21}>\frac{1}{22}$ nên $\frac{1}{12}-\frac{1}{22}>0,\frac{1}{13}-\frac{1}{22}>0,\mathrm{...},\frac{1}{21}-\frac{1}{22}>0,\frac{1}{22}-\frac{1}{22}=0$

Suy ra $A>\frac{1}{2}$