Question Number 174514 by Best1 last updated on 03/Aug/22
$${a}+{b}+{c}=\mathrm{1} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{2} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{3} \\ $$$${then}\:{a}^{\mathrm{5}} +{b}^{\mathrm{5}} +{c}^{\mathrm{5}} \:? \\ $$
Commented by mr W last updated on 03/Aug/22
$${see}\:{Q}\mathrm{74970} \\ $$
Answered by RedMath last updated on 03/Aug/22
$${no}\:{idea}\:{lol} \\ $$
Commented by Best1 last updated on 03/Aug/22
$${sorry} \\ $$
Answered by MJS_new last updated on 03/Aug/22
$$\mathrm{1}.\:{a}+{b}+{c}=\alpha \\ $$$$\mathrm{2}.\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\beta \\ $$$$\mathrm{3}.{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\gamma \\ $$$$======================== \\ $$$$\mathrm{let}\:{a}={u}−\sqrt{{v}}\wedge{b}={u}+\sqrt{{v}} \\ $$$$\Rightarrow \\ $$$$\mathrm{1}.\:{c}=\alpha−\mathrm{2}{u} \\ $$$$\mathrm{2}.\:\mathrm{6}{u}^{\mathrm{2}} −\mathrm{4}\alpha{u}+\mathrm{2}{v}+\alpha^{\mathrm{2}} −\beta=\mathrm{0} \\ $$$$\mathrm{3}.\:−\mathrm{6}{u}^{\mathrm{3}} +\mathrm{12}\alpha{u}^{\mathrm{2}} +\mathrm{6}{u}\left({v}−\alpha^{\mathrm{2}} \right)+\alpha^{\mathrm{3}} −\gamma=\mathrm{0} \\ $$$$======================== \\ $$$$\mathrm{2}.\:{v}=−\mathrm{3}{u}^{\mathrm{2}} +\mathrm{2}\alpha{u}−\frac{\alpha^{\mathrm{2}} −\beta}{\mathrm{2}} \\ $$$$\mathrm{3}.\:{u}^{\mathrm{3}} −\alpha{u}^{\mathrm{2}} +\frac{\mathrm{3}\alpha^{\mathrm{2}} −\beta}{\mathrm{8}}{u}=\frac{\alpha^{\mathrm{3}} −\gamma}{\mathrm{24}} \\ $$$$======================== \\ $$$${a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{c}^{\mathrm{4}} = \\ $$$$=−\mathrm{32}\alpha\left({u}^{\mathrm{3}} −\alpha{u}^{\mathrm{2}} +\frac{\mathrm{3}\alpha^{\mathrm{2}} −\beta}{\mathrm{8}}{u}\right)+\frac{\mathrm{3}\alpha^{\mathrm{4}} −\mathrm{2}\alpha^{\mathrm{2}} \beta+\beta^{\mathrm{2}} }{\mathrm{2}}= \\ $$$$=\frac{\alpha^{\mathrm{4}} −\mathrm{6}\alpha^{\mathrm{2}} \beta+\mathrm{8}\alpha\gamma+\mathrm{3}\beta^{\mathrm{2}} }{\mathrm{6}} \\ $$$$======================== \\ $$$${a}^{\mathrm{5}} +{b}^{\mathrm{5}} +{c}^{\mathrm{5}} = \\ $$$$=−\mathrm{20}\left(\alpha^{\mathrm{2}} +\beta\right)\left({u}^{\mathrm{3}} −\alpha{u}^{\mathrm{2}} +\frac{\mathrm{3}\alpha^{\mathrm{2}} −\beta}{\mathrm{8}}{u}\right)+\alpha^{\mathrm{5}} = \\ $$$$=\frac{\alpha^{\mathrm{5}} −\mathrm{5}\left(\alpha^{\mathrm{3}} \beta−\alpha^{\mathrm{2}} \gamma−\mathrm{5}{l}\beta\gamma\right)}{\mathrm{6}} \\ $$$$======================== \\ $$
Commented by Tawa11 last updated on 03/Aug/22
$$\mathrm{Great}\:\mathrm{sirs}. \\ $$
Answered by behi834171 last updated on 03/Aug/22
$${a}^{\mathrm{2}} +{ab}+{ac}={a} \\ $$$${ab}+{b}^{\mathrm{2}} +{bc}={b} \\ $$$${ac}+{bc}+{c}^{\mathrm{2}} ={c}\Rightarrow\mathrm{2}\Sigma{ab}=\Sigma{a}−\Sigma{a}^{\mathrm{2}} =\mathrm{1}−\mathrm{2}=−\mathrm{1} \\ $$$$\Rightarrow\Sigma{ab}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${a}^{\mathrm{3}} +{a}\left({b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)=\mathrm{2}{a} \\ $$$${b}\left({a}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)+{b}^{\mathrm{3}} =\mathrm{2}{b} \\ $$$${c}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)+{c}^{\mathrm{3}} =\mathrm{2}{c}\Rightarrow \\ $$$$\Rightarrow\Sigma{ab}\left({a}+{b}\right)=\mathrm{2}\Sigma{a}−\Sigma{a}^{\mathrm{3}} =\mathrm{2}−\mathrm{3}=−\mathrm{1} \\ $$$$\Rightarrow\Sigma{ab}\left(\mathrm{1}−{c}\right)=−\mathrm{1}\Rightarrow\Sigma{ab}−\mathrm{3}{abc}=−\mathrm{1} \\ $$$$\Rightarrow−\mathrm{3}{abc}=−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}=−\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow\boldsymbol{{abc}}=\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$\boldsymbol{{so}}:\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}},\:{are}\:{the}\:{roots}\:{of}\:{equation}: \\ $$$$\:\:\:\:\:\boldsymbol{{z}}^{\mathrm{3}} −\boldsymbol{{z}}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}−\frac{\mathrm{1}}{\mathrm{6}}=\mathrm{0} \\ $$$$\boldsymbol{{z}}^{\mathrm{5}} =\boldsymbol{{z}}^{\mathrm{3}} .\boldsymbol{{z}}^{\mathrm{2}} =\boldsymbol{{z}}^{\mathrm{2}} \left(\boldsymbol{{z}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}+\frac{\mathrm{1}}{\mathrm{6}}\right)=\boldsymbol{{z}}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{6}}\boldsymbol{{z}}^{\mathrm{2}} = \\ $$$$=\boldsymbol{{z}}\left(\boldsymbol{{z}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}+\frac{\mathrm{1}}{\mathrm{6}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{z}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}+\frac{\mathrm{1}}{\mathrm{6}}\right)+\frac{\mathrm{1}}{\mathrm{6}}\boldsymbol{{z}}^{\mathrm{2}} = \\ $$$$=\boldsymbol{{z}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}+\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{2}}{z}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{6}}{z}+\frac{\mathrm{1}}{\mathrm{2}}\boldsymbol{{z}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{4}}{z}+\frac{\mathrm{1}}{\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{6}}\boldsymbol{{z}}^{\mathrm{2}} = \\ $$$$=\frac{\mathrm{13}}{\mathrm{6}}\boldsymbol{{z}}^{\mathrm{2}} +\frac{\mathrm{11}}{\mathrm{12}}\boldsymbol{{z}}+\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\Rightarrow\Sigma\boldsymbol{{a}}^{\mathrm{5}} =\frac{\mathrm{13}}{\mathrm{6}}\Sigma\boldsymbol{{a}}^{\mathrm{2}} +\frac{\mathrm{11}}{\mathrm{12}}\Sigma\boldsymbol{{a}}+\frac{\mathrm{3}}{\mathrm{4}}= \\ $$$$=\frac{\mathrm{13}}{\mathrm{6}}×\mathrm{2}+\frac{\mathrm{11}}{\mathrm{12}}×\mathrm{1}+\frac{\mathrm{3}}{\mathrm{4}}=\mathrm{6}\:\:\:\:.\blacksquare \\ $$$$……………………………….. \\ $$$${by}\:{using}\:{symbols}\:{of}\:{dear}\:{mr}:{MJS\_new} \\ $$$$\Sigma\boldsymbol{{ab}}=\frac{\Sigma\boldsymbol{{a}}−\Sigma\boldsymbol{{a}}^{\mathrm{2}} }{\mathrm{2}}=\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}} \\ $$$$\Sigma\boldsymbol{{ab}}−\mathrm{3}\boldsymbol{{abc}}=\mathrm{2}\Sigma\boldsymbol{{a}}−\Sigma\boldsymbol{{a}}^{\mathrm{3}} \Rightarrow \\ $$$$\boldsymbol{{abc}}=−\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{2}\boldsymbol{\alpha}−\boldsymbol{\gamma}−\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\right)=−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}} \\ $$$$\Rightarrow\boldsymbol{{z}}^{\mathrm{3}} −\boldsymbol{\alpha{z}}^{\mathrm{2}} +\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\boldsymbol{{z}}+\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}=\mathrm{0} \\ $$$$\Rightarrow\boldsymbol{{z}}^{\mathrm{5}} =\boldsymbol{{z}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{3}} =\boldsymbol{{z}}^{\mathrm{2}} \left(\boldsymbol{\alpha{z}}^{\mathrm{2}} −\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\boldsymbol{{z}}−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\right)= \\ $$$$=\boldsymbol{\alpha{z}}\left(\boldsymbol{\alpha{z}}^{\mathrm{2}} −\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\boldsymbol{{z}}−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\right)−\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\left(\boldsymbol{\alpha{z}}^{\mathrm{2}} −\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\boldsymbol{{z}}−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\right)−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\boldsymbol{{z}}^{\mathrm{2}} = \\ $$$$=\boldsymbol{\alpha}^{\mathrm{2}} \left(\boldsymbol{\alpha{z}}^{\mathrm{2}} −\frac{\boldsymbol{\alpha}−\boldsymbol{\beta}}{\mathrm{2}}\boldsymbol{{z}}−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\right)−\frac{\boldsymbol{\alpha}\left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)}{\mathrm{2}}\boldsymbol{{z}}^{\mathrm{2}} −\frac{\boldsymbol{\alpha}\left(\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}\right)}{\mathrm{6}}\boldsymbol{{z}}− \\ $$$$+\frac{\boldsymbol{\alpha}\left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)}{\mathrm{2}}\boldsymbol{{z}}^{\mathrm{2}} +\frac{\left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)^{\mathrm{2}} }{\mathrm{4}}\boldsymbol{{z}}+\frac{\left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)\left(\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}\right)}{\mathrm{12}}−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\boldsymbol{{z}}^{\mathrm{2}} = \\ $$$$=\left[\boldsymbol{\alpha}^{\mathrm{3}} −\frac{\boldsymbol{\alpha}\left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)}{\mathrm{2}}−\frac{\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}}\right]\boldsymbol{{z}}^{\mathrm{2}} −\left[\frac{\boldsymbol{\alpha}^{\mathrm{2}} \left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)}{\mathrm{2}}+\frac{\boldsymbol{\alpha}\left(\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}\right)}{\mathrm{6}}−\frac{\left(\boldsymbol{\alpha}−\boldsymbol{\beta}\right)^{\mathrm{2}} }{\mathrm{4}}\right].\boldsymbol{{z}}− \\ $$$$−\frac{\left(\mathrm{2}\boldsymbol{\alpha}^{\mathrm{2}} −\boldsymbol{\alpha}+\boldsymbol{\beta}\right)\left(\mathrm{3}\boldsymbol{\alpha}+\boldsymbol{\beta}−\mathrm{2}\boldsymbol{\gamma}\right)}{\mathrm{12}} \\ $$$$=\boldsymbol{{A}}.\boldsymbol{{z}}^{\mathrm{2}} +\boldsymbol{{B}}.\boldsymbol{{z}}+\boldsymbol{{C}} \\ $$$$\boldsymbol{{A}}=\frac{\mathrm{6}\boldsymbol{\alpha}^{\mathrm{3}} −\mathrm{3}\boldsymbol{\alpha}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\alpha\beta}−\mathrm{3}\boldsymbol{\alpha}−\boldsymbol{\beta}+\mathrm{2}\boldsymbol{\gamma}}{\mathrm{6}} \\ $$$$\boldsymbol{{B}}=−\frac{\mathrm{6}\boldsymbol{\alpha}^{\mathrm{3}} −\mathrm{6}\boldsymbol{\alpha}^{\mathrm{2}} \boldsymbol{\beta}+\mathrm{3}\boldsymbol{\alpha}^{\mathrm{2}} +\mathrm{8}\boldsymbol{\alpha\beta}−\mathrm{4}\boldsymbol{\alpha\gamma}−\mathrm{3}\boldsymbol{\beta}^{\mathrm{2}} }{\mathrm{12}} \\ $$$$\boldsymbol{{C}}=−\frac{\mathrm{6}\boldsymbol{\alpha}^{\mathrm{3}} +\mathrm{2}\boldsymbol{\alpha}^{\mathrm{2}} \boldsymbol{\beta}−\mathrm{4}\boldsymbol{\alpha}^{\mathrm{2}} \boldsymbol{\gamma}−\mathrm{3}\boldsymbol{\alpha}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\alpha\beta}+\mathrm{2}\boldsymbol{\alpha\gamma}+\boldsymbol{\beta}^{\mathrm{2}} −\mathrm{2}\boldsymbol{\beta\gamma}}{\mathrm{12}} \\ $$$$\Rightarrow\Sigma\boldsymbol{{a}}^{\mathrm{5}} =\boldsymbol{{A}}.\Sigma\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{B}}.\Sigma\boldsymbol{{a}}+\mathrm{3}\boldsymbol{{C}} \\ $$
Commented by Best1 last updated on 03/Aug/22
$$ \\ $$$${wawww}\:{thanks}\:{guys} \\ $$
Commented by Tawa11 last updated on 03/Aug/22
$$\mathrm{Great}\:\mathrm{sirs} \\ $$