Question Number 122788 by 676597498 last updated on 19/Nov/20
Answered by mr W last updated on 20/Nov/20
![0<(1/2)+(1/(1000))<1⇒[(1/2)+(1/(1000))]=0 ... 0<(1/2)+((499)/(1000))<1⇒[(1/2)+((499)/(1000))]=0 (1/2)+((500)/(1000))=1⇒[(1/2)+((500)/(1000))]=1 1<(1/2)+((501)/(1000))<2⇒[(1/2)+((501)/(1000))]=1 ... 1<(1/2)+((999)/(1000))<2⇒[(1/2)+((999)/(1000))]=1 ⇒sum=999−499=500](https://www.tinkutara.com/question/Q122790.png)
$$\mathrm{0}<\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{1000}}<\mathrm{1}\Rightarrow\left[\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{1000}}\right]=\mathrm{0} \\ $$$$… \\ $$$$\mathrm{0}<\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{499}}{\mathrm{1000}}<\mathrm{1}\Rightarrow\left[\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{499}}{\mathrm{1000}}\right]=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{500}}{\mathrm{1000}}=\mathrm{1}\Rightarrow\left[\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{500}}{\mathrm{1000}}\right]=\mathrm{1} \\ $$$$\mathrm{1}<\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{501}}{\mathrm{1000}}<\mathrm{2}\Rightarrow\left[\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{501}}{\mathrm{1000}}\right]=\mathrm{1} \\ $$$$… \\ $$$$\mathrm{1}<\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{999}}{\mathrm{1000}}<\mathrm{2}\Rightarrow\left[\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{999}}{\mathrm{1000}}\right]=\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow{sum}=\mathrm{999}−\mathrm{499}=\mathrm{500} \\ $$