Category: Arithmetic

Question Number 62570 by aliesam last updated on 23/Jun/19 Commented by mathmax by abdo last updated on 23/Jun/19 $${we}\:{have}\:{S}\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{sin}\left({nx}\right)}{\mathrm{2}^{{n}−\mathrm{1}} }\:=\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{sin}\left({nx}\right)}{\mathrm{2}^{{n}} }\:=\mathrm{2}\:{Im}\left(\sum_{{n}=\mathrm{1}}…

Question Number 128012 by mathocean1 last updated on 03/Jan/21 $${show}\:{that}\:\forall\:{n}\:\in\:\mathbb{N},\:\mathrm{12}\:{divise}\:{n}^{\mathrm{2}} \left({n}^{\mathrm{4}} −\mathrm{1}\right) \\ $$ Answered by MJS_new last updated on 03/Jan/21 $${n}=\mathrm{2}{k}\:\Rightarrow\:\mathrm{2}\mid{n}\:\Rightarrow\:\mathrm{4}\mid{n}^{\mathrm{2}} \\ $$$${n}=\mathrm{2}{k}+\mathrm{1}\:\Rightarrow\:{n}^{\mathrm{4}} −\mathrm{1}=\mathrm{8}{k}\left({k}−\mathrm{1}\right)\left(\mathrm{2}{k}^{\mathrm{2}}…

Question Number 62322 by aliesam last updated on 19/Jun/19 Commented by arcana last updated on 19/Jun/19 $$\beta_{\mathrm{1}} =\alpha+\alpha^{\mathrm{6}} \\ $$$$\beta_{\mathrm{2}} =\left(\alpha+\alpha^{\mathrm{6}} \right)^{\mathrm{2}} −\mathrm{2}=\alpha^{\mathrm{2}} +\mathrm{2}\alpha^{\mathrm{7}} +\alpha^{\mathrm{12}}…

Question Number 62263 by Tawa1 last updated on 18/Jun/19 Answered by behi83417@gmail.com last updated on 19/Jun/19 $$\mathrm{x}+\sqrt{\mathrm{x}−\mathrm{2}}=\mathrm{a}\Rightarrow\frac{\mathrm{2}}{\mathrm{a}}=\frac{\mathrm{1}}{\mathrm{x}+\sqrt{\mathrm{x}−\mathrm{2}}}=\frac{\mathrm{x}−\sqrt{\mathrm{x}−\mathrm{2}}}{\mathrm{1}} \\ $$$$\frac{\mathrm{2}}{\mathrm{a}}+\sqrt{\mathrm{a}}=\mathrm{3}\Rightarrow\mathrm{a}\sqrt{\mathrm{a}}−\mathrm{3a}+\mathrm{2}=\mathrm{0} \\ $$$$\sqrt{\mathrm{a}}=\mathrm{t}\Rightarrow\mathrm{t}^{\mathrm{3}} −\mathrm{3t}^{\mathrm{2}} +\mathrm{2}=\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{t}−\mathrm{1}\right)\left(\mathrm{t}^{\mathrm{2}}…

Question Number 127504 by Snail last updated on 30/Dec/20 Terms of Service Privacy Policy Contact: info@tinkutara.com