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Question Number 128462 by Ari last updated on 07/Jan/21 $$\mathrm{coukd}\:\mathrm{you}\:\mathrm{prove}\:\mathrm{please}\:\mathrm{that}\:\sqrt{\mathrm{11}\:}\mathrm{is}\:\mathrm{not}\:\mathrm{racional}\:\mathrm{number} \\ $$ Answered by Dwaipayan Shikari last updated on 07/Jan/21 $$\sqrt{\mathrm{11}}=\mathrm{3}+\sqrt{\mathrm{11}}−\mathrm{3}=\mathrm{3}+\frac{\mathrm{2}}{\:\sqrt{\mathrm{11}}+\mathrm{3}}=\mathrm{3}+\frac{\mathrm{2}}{\mathrm{3}+\mathrm{3}+\frac{\mathrm{2}}{\mathrm{3}+\mathrm{3}+\frac{\mathrm{2}}{\mathrm{3}+\mathrm{3}+\frac{\mathrm{2}}{\mathrm{3}+\mathrm{3}…}}}} \\ $$$$=\mathrm{3}+\frac{\mathrm{2}}{\mathrm{6}+\frac{\mathrm{2}}{\mathrm{6}+\frac{\mathrm{2}}{\mathrm{6}+\frac{\mathrm{2}}{\mathrm{6}+\frac{\mathrm{2}}{\mathrm{6}+…}}}}}\:\:\:\:\: \\ $$$${It}\:{is}\:{an}\:{Infinte}\:{continued}\:{fraction}\:.{So}\:{it}\:{can}\:{never}\:{be}\:{Rational}…
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Question Number 62570 by aliesam last updated on 23/Jun/19 Commented by mathmax by abdo last updated on 23/Jun/19 $${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 128038 by Walt123 last updated on 03/Jan/21 $$ \\ $$Resolva a equação abaixo: $$ \\ $$$$\mathrm{5}^{\mathrm{x}} .\mathrm{16}^{\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}}} =\mathrm{100} \\ $$ Answered by MJS_new…
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 $${n}=\mathrm{2}{k}\:\Rightarrow\:\mathrm{2}\mid{n}\:\Rightarrow\:\mathrm{4}\mid{n}^{\mathrm{2}} \\ $$$${n}=\mathrm{2}{k}+\mathrm{1}\:\Rightarrow\:{n}^{\mathrm{4}} −\mathrm{1}=\mathrm{8}{k}\left({k}−\mathrm{1}\right)\left(\mathrm{2}{k}^{\mathrm{2}}…
Question Number 127865 by Ar Brandon last updated on 02/Jan/21 Answered by mathmax by abdo last updated on 03/Jan/21 $$\left.\mathrm{a}\right)\mathrm{wehave}\:\mathrm{u}_{\mathrm{n}+\mathrm{1}\:} =\mathrm{f}\left(\mathrm{u}_{\mathrm{n}} \right)\:\mathrm{with}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}−\mathrm{x}^{\mathrm{2}} \:\:\mathrm{f}\:\mathrm{is}\:\mathrm{contnue}\: \\ $$$$\mathrm{the}\:\mathrm{fix}\:\mathrm{point}\:\:\mathrm{verify}\:\mathrm{x}=\mathrm{x}−\mathrm{x}^{\mathrm{2}}…
Question Number 62322 by aliesam last updated on 19/Jun/19 Commented by arcana last updated on 19/Jun/19 $$\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}}…