Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 121989 by mlj5 last updated on 13/Nov/20

$$determinerleplusgrandentierdentelque$$$$\frac{1}{\sqrt{2}+\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+...\frac{1}{\sqrt{n+1}+\sqrt{n}}⩽9$$

Answered by Ar Brandon last updated on 13/Nov/20

$${\mathrm{S}}_{\mathrm{n}}=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{1}{\sqrt{\mathrm{k}+1}+\sqrt{\mathrm{k}}}=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{\sqrt{\mathrm{k}+1}-\sqrt{\mathrm{k}}}{1}$$$$=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}[\sqrt{\mathrm{k}+1}-\sqrt{\mathrm{k}}]=(\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+\cdot \cdot \cdot$$$$+(\sqrt{\mathrm{n}}-\sqrt{\mathrm{n}-1})+(\sqrt{\mathrm{n}+1}-\sqrt{\mathrm{n}})$$$$\mathrm{Par}\mathrm{t}\acute{\mathrm{e}l}\acute{\mathrm{e}scopage}\mathrm{nous}\mathrm{avons}-\sqrt{1}+\sqrt{\mathrm{n}+1}$$$$\sqrt{\mathrm{n}+1}-1⩽9\Rightarrow \sqrt{\mathrm{n}+1}⩽10$$$$\Rightarrow \mathrm{n}+1⩽100\Rightarrow \mathrm{n}⩽99$$$${\mathrm{D}}^{\prime }\mathrm{o}\stackrel{`}{\mathrm{u}}\mathrm{le}\mathrm{plus}\mathrm{grand}\mathrm{entier}\mathrm{est}\mathrm{n}=99$$

Answered by Dwaipayan Shikari last updated on 13/Nov/20

$$\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+...n$$$$\sqrt{2}-1+\sqrt{3}-\sqrt{2}+....+\sqrt{n+1}-\sqrt{n}$$$$\sqrt{n+1}-1$$$$n=99$$$$\sqrt{100}-1=9$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com