cho C=1/(1+√2)+1/(√2+√3)+...+1/(√99+√100) chứng tỏ C là một số nguyên

$\displaystyle \begin{array}{{>{\displaystyle}l}} C=\frac{1}{1+\sqrt{2}} +\frac{1}{\sqrt{2} +\sqrt{3}} +..+\frac{1}{\sqrt{99} +\sqrt{100}} \ \\ C=\frac{\sqrt{2} -1}{\left(\sqrt{2} -1\right)\left(\sqrt{2} +1\right)} +\frac{\sqrt{3} -\sqrt{2}}{\left(\sqrt{2} +\sqrt{3}\right)\left(\sqrt{3} -\sqrt{2}\right)} +...+\frac{\sqrt{100} -\sqrt{99}}{\left(\sqrt{100} -\sqrt{99}\right)\left(\sqrt{100} +\sqrt{99}\right)}\\ C=\frac{\sqrt{2} -1}{2-1} +\frac{\sqrt{3} -\sqrt{2}}{3-2} +..+\frac{\sqrt{100} -\sqrt{99}}{100-99}\\ C=\sqrt{2} -1+\sqrt{3} -\sqrt{2} +...+\sqrt{100} -\sqrt{99}\\ C=-1+\sqrt{100}\\ C=-1+10=9\ là\ số\ nguyên\ \\ Vậy\ C\ là\ một\ số\ nguyên\ \end{array}$

`C=1/(1+\sqrt{2})+1/(\sqrt{2}+\sqrt{3})+...+1/(\sqrt{99}+\sqrt{100})`

`C=(\sqrt{2}-1)/((1+\sqrt{2})(\sqrt{2}-1))+(\sqrt{3}-\sqrt{2})/((\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2}))+...+(\sqrt{99}-\sqrt{100})/((\sqrt{99}+\sqrt{100})(\sqrt{100}-\sqrt{99})`

`C=(\sqrt{2}-1)/(2-1)+(\sqrt{3}-\sqrt{2})/(2-1)+...+(\sqrt{100}-\sqrt{99})/(100-99)`

`C=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{100}-\sqrt{99}`

`C=-1+\sqrt{100}`

`C=-1+10`

`C=9∈Z`

Vậy `C=1/(1+\sqrt{2})+1/(\sqrt{2}+\sqrt{3})+...+1/(\sqrt{99}+\sqrt{100})` là một số nguyên `(đpcm)`