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Category: Algebra

Question-196885

Question Number 196885 by hardmath last updated on 02/Sep/23 Answered by MM42 last updated on 02/Sep/23 $$\mathrm{2}{cosx}=\sqrt{\mathrm{2}+\mathrm{2}{cos}\mathrm{2}{x}}=\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+{cos}\mathrm{4}{x}}} \\ $$$$=\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+{cos}\mathrm{8}{x}}}}=….=\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+…}}} \\ $$$${let}\:\:{x}=\mathrm{0}\Rightarrow\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+…}}}=\mathrm{2}\:\checkmark \\ $$$$ \\ $$…

Let-be-a-positive-Root-of-x-2-2023x-1-Define-a-sequence-i-such-That-0-1-n-1-n-find-The-Remainder-When-2023-is-divided-by-2-

Question Number 196872 by York12 last updated on 02/Sep/23 $${Let}\:\xi\:{be}\:{a}\:{positive}\:{Root}\:{of}\:{x}^{\mathrm{2}} −\mathrm{2023}{x}−\mathrm{1} \\ $$$${Define}\:{a}\:{sequence}\:\varphi_{{i}} \:{such}\:{That}\:\varphi_{\mathrm{0}} =\mathrm{1} \\ $$$$\varphi_{{n}+\mathrm{1}} =\lfloor\varphi_{{n}} \xi\rfloor,\:{find}\:{The}\:{Remainder}\:{When}\:\varphi_{\mathrm{2023}\:} {is}\:{divided}\:{by}\:\sqrt{\varphi_{\mathrm{2}} } \\ $$ Answered by…

let-b-i-a-i-gt-0-where-i-1-2-3-n-amp-i-1-n-b-i-Prove-that-b-1-b-2-b-1-b-2-a-1-a-2-b-1-b-3-b-1-b-3-a-1-a-3-b-2-b-3-b-2-b-3-

Question Number 196870 by York12 last updated on 02/Sep/23 $${let}\:{b}_{{i}} \wedge\:{a}_{{i}} >\mathrm{0}\:{where}\:{i}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,{n}\right\}\&\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left({b}_{{i}} \right)=\lambda\:{Prove}\:{that} \\ $$$$\frac{\lambda−\left({b}_{\mathrm{1}} +{b}_{\mathrm{2}} \right)}{\left({b}_{\mathrm{1}} +{b}_{\mathrm{2}} \right)}\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} \right)+\frac{\lambda−\left({b}_{\mathrm{1}} +{b}_{\mathrm{3}} \right)}{\left({b}_{\mathrm{1}}…

6-5-9-7-16-1-4-10-

Question Number 196815 by mathlove last updated on 01/Sep/23 $$!\mathrm{6}×\frac{\sqrt[{\mathrm{4}}]{!\mathrm{5}+\mathrm{9}!!!!!+\mathrm{7}!!!−\mathrm{16}}}{!\mathrm{10}}=? \\ $$ Answered by Tokugami last updated on 01/Sep/23 $$\mathrm{9}!!!!!=\mathrm{9}×\left(\mathrm{9}−\mathrm{5}\right)=\mathrm{36} \\ $$$$\mathrm{7}!!!=\mathrm{7}×\left(\mathrm{7}−\mathrm{3}\right)×\left(\mathrm{7}−\mathrm{6}\right)=\mathrm{28} \\ $$$$!{n}=\left[\frac{{n}!}{{e}}\right] \\…

Question-196714

Question Number 196714 by mr W last updated on 30/Aug/23 Answered by som(math1967) last updated on 30/Aug/23 $$\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}=\frac{\mathrm{1}}{{a}+{b}+{c}} \\ $$$$\Rightarrow\left({a}+{b}+{c}\right)\left({ab}+{bc}+{ca}\right)−{abc}=\mathrm{0} \\ $$$$\Rightarrow\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right)=\mathrm{0} \\ $$$$\:{a}+{b}=\mathrm{0}\Rightarrow{a}=−{b}\Rightarrow{c}=\mathrm{2022} \\…