Tính P= 1/2+ 1/3+ 1/4 + ...+ 1/2012/ 2011/1+ 2010/2+ 2009/3+ ...+ 1/2011

Trả lời

Đặt $B=\frac{\text{2011}}{\text{1}}\text{ + }\frac{\text{2010}}{\text{2}}\text{ + }\frac{\text{2009}}{\text{3}}\text{ + }\mathrm{...}\text{ + }\frac{\text{1}}{\text{2011}}$

Công 1 vào ta được

$B=\left(\frac{\text{2012}}{\text{1}}\text{ + }\frac{\text{2012}}{\text{2}}\text{ + }\frac{\text{2012}}{\text{3}}\text{ + }\mathrm{...}\text{ + }\frac{2012}{\text{2011}}+\frac{2012}{2012}\right)-2012$

$\Rightarrow \text{B = 2012}\left(\frac{\text{1}}{\text{2}}\text{ + }\frac{\text{1}}{\text{3}}\text{ + }\mathrm{...}\text{ + }\frac{\text{1}}{\text{2011}}\text{ + }\frac{\text{1}}{\text{2012}}\right)\text{ - 2012 + }\frac{\text{2012}}{\text{1}}$

Thay vào P ta được

$P=\frac{\frac{\text{1}}{\text{2}}\text{ + }\frac{\text{1}}{\text{3}}\text{ + }\mathrm{...}\text{ + }\frac{\text{1}}{\text{2012}}}{2012\left(\frac{\text{1}}{\text{2}}\text{ + }\frac{\text{1}}{\text{3}}\text{ + }\mathrm{...}\text{ + }\frac{\text{1}}{\text{2012}}\right)}$

$\Rightarrow P=\frac{1}{2012}$