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Question-185361

Question Number 185361 by yaslm last updated on 20/Jan/23 Answered by Rasheed.Sindhi last updated on 20/Jan/23 $$\mathrm{2}{x}+{y}>\mathrm{5}….\left({i}\right) \\ $$$$−\mathrm{1}>{y}……\left({ii}\right) \\ $$$$\left({i}\right)+\left({ii}\right) \\ $$$$\mathrm{2}{x}+{y}−\mathrm{1}>{y}+\mathrm{5} \\ $$$$\mathrm{2}{x}−\mathrm{1}>\mathrm{5}…

Question-185354

Question Number 185354 by Ml last updated on 20/Jan/23 Answered by HeferH last updated on 20/Jan/23 $$\:\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\:−\:\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\:=\:{k} \\ $$$$\:\mathrm{2}\:+\:\sqrt{\mathrm{3}}\:+\:\mathrm{2}\:−\:\sqrt{\mathrm{3}}\:−\:\mathrm{2}\left(\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{3}}}\:\centerdot\:\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\right)={k}^{\mathrm{2}} \\ $$$$\:\:\mathrm{4}\:−\mathrm{2}\left(\sqrt{\mathrm{2}^{\mathrm{2}} −\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} }\right)=\:{k}^{\mathrm{2}} \\ $$$$\:\sqrt{\mathrm{2}}\:=\:{k}\:…

Question-119803

Question Number 119803 by repentence last updated on 27/Oct/20 Answered by Dwaipayan Shikari last updated on 27/Oct/20 $${T}_{{n}} =\sqrt{\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }} \\ $$$$=\sqrt{\frac{\left({n}^{\mathrm{4}} +\mathrm{2}{n}^{\mathrm{3}} +{n}^{\mathrm{2}}…

lim-k-i-1-k-1-i-1-i-

Question Number 185329 by liuxinnan last updated on 20/Jan/23 $${lim}_{{k}\rightarrow+\infty} \underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{i}−\mathrm{1}}+\sqrt{{i}}}= \\ $$ Answered by aleks041103 last updated on 20/Jan/23 $$\sqrt{{i}−\mathrm{1}}<\sqrt{{i}}\Rightarrow\sqrt{{i}−\mathrm{1}}+\sqrt{{i}}<\mathrm{2}\sqrt{{i}}\leqslant\mathrm{2}{i} \\ $$$$\Rightarrow\underset{{i}=\mathrm{1}}…

lim-k-i-1-k-1-i-

Question Number 185328 by liuxinnan last updated on 20/Jan/23 $${lim}_{{k}\rightarrow+\infty} \underset{{i}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{{i}}}= \\ $$ Answered by aleks041103 last updated on 20/Jan/23 $${since}\:\sqrt{{i}}\leqslant{i}\Rightarrow\frac{\mathrm{1}}{\:\sqrt{{i}}}\geqslant\frac{\mathrm{1}}{{i}} \\ $$$$\Rightarrow\underset{{i}=\mathrm{1}}…

Question-185325

Question Number 185325 by Ml last updated on 20/Jan/23 Answered by aleks041103 last updated on 20/Jan/23 $$\mathrm{14}+\mathrm{8}\sqrt{\mathrm{3}}=\mathrm{14}+\mathrm{2}.\mathrm{2}\sqrt{\mathrm{2}}.\sqrt{\mathrm{6}}= \\ $$$$=\left(\sqrt{\mathrm{6}}\right)^{\mathrm{2}} +\left(\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{2}\left(\mathrm{2}\sqrt{\mathrm{2}}\right)\left(\sqrt{\mathrm{6}}\right)= \\ $$$$=\left(\sqrt{\mathrm{6}}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\sqrt{\mathrm{14}+\mathrm{8}\sqrt{\mathrm{3}}}=\sqrt{\left(\sqrt{\mathrm{6}}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{\mathrm{2}}…