Đố. Đố em tính nhanh được tổng

$\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}$.

Giải

$\frac{1}{x\left(x+1\right)}=\frac{1}{x}-\frac{1}{x+1}$

$\frac{1}{\left(x+1\right)\left(x+2\right)}=\frac{1}{x+1}-\frac{1}{x+2}$

...........................................................

$\frac{1}{\left(x+5\right)\left(x+6\right)}=\frac{1}{x+5}-\frac{1}{x+6}$

Do đó:

$$\begin{gathered} \frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)} \\ =\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+.......+\frac{1}{x+5}-\frac{1}{x+6} \\ =\frac{1}{x}-\frac{1}{x+6}=\frac{x+6-x}{x(x+6)}=\frac{6}{x(x+6)} \end{gathered}$$