Question Number 204129 by hardmath last updated on 06/Feb/24
$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{x}\:,\:\mathrm{y}\:\in\:\mathbb{R} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{23} \\ $$$$\mathrm{ax}\:+\:\mathrm{by}\:=\:\mathrm{79} \\ $$$$\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{by}^{\mathrm{2}} \:=\:\mathrm{217} \\ $$$$\mathrm{ax}^{\mathrm{3}} \:+\:\mathrm{by}^{\mathrm{3}} \:=\:\mathrm{661} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{ax}^{\mathrm{4}} \:+\:\mathrm{by}^{\mathrm{4}} \:=\:? \\ $$
Answered by AST last updated on 06/Feb/24
$${x}\left({ax}+{by}\right)=\mathrm{79}{x}\Rightarrow{ax}^{\mathrm{2}} +{bxy}=\mathrm{79}{x} \\ $$$${axy}+{by}^{\mathrm{2}} =\mathrm{79}{y}\Rightarrow{ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +{xy}\left({a}+{b}\right)=\mathrm{79}\left({x}+{y}\right) \\ $$$$\Rightarrow\mathrm{217}=\mathrm{79}\left({x}+{y}\right)−\mathrm{23}{xy}…\left({i}\right) \\ $$$${ax}^{\mathrm{3}} +{bxy}^{\mathrm{2}} =\mathrm{217}{x};\:{ax}^{\mathrm{2}} {y}+{by}^{\mathrm{3}} =\mathrm{217}{y} \\ $$$$\mathrm{661}+{xy}\left({by}+{ax}\right)=\mathrm{217}\left({x}+{y}\right) \\ $$$$\Rightarrow\mathrm{661}=\mathrm{217}\left({x}+{y}\right)−\mathrm{79}{xy}…\left({ii}\right) \\ $$$${ax}^{\mathrm{4}} +{by}^{\mathrm{4}} ={x}\left({ax}^{\mathrm{3}} +{by}^{\mathrm{3}} \right)+{y}\left({ax}^{\mathrm{3}} +{by}^{\mathrm{3}} \right)−{xy}\left({by}^{\mathrm{2}} +{ax}^{\mathrm{2}} \right) \\ $$$$=\mathrm{661}{x}+\mathrm{661}{y}−\mathrm{217}{xy} \\ $$$$\mathrm{23}\left({ii}\right)−\mathrm{79}\left({i}\right)\Rightarrow−\mathrm{1250}\left({x}+{y}\right)=−\mathrm{1940}\Rightarrow{x}+{y}=\frac{\mathrm{194}}{\mathrm{125}} \\ $$$$\mathrm{79}\left({ii}\right)−\mathrm{217}\left({i}\right)\Rightarrow\mathrm{5130}=−\mathrm{1250}{xy}\Rightarrow{xy}=−\frac{\mathrm{513}}{\mathrm{125}} \\ $$$$\Rightarrow{ax}^{\mathrm{4}} +{by}^{\mathrm{4}} =\mathrm{661}×\frac{\mathrm{194}}{\mathrm{125}}+\frac{\mathrm{217}×\mathrm{513}}{\mathrm{125}}=\frac{\mathrm{47911}}{\mathrm{25}} \\ $$
Answered by mr W last updated on 06/Feb/24
$$\left({ax}+{by}\right)\left({x}+{y}\right)={ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +\left({a}+{b}\right){xy} \\ $$$$\Rightarrow\mathrm{79}\left({x}+{y}\right)=\mathrm{217}+\mathrm{23}{xy}\:\:\:…\left({i}\right) \\ $$$$\left({ax}^{\mathrm{2}} +{by}^{\mathrm{2}} \right)\left({x}+{y}\right)={ax}^{\mathrm{3}} +{by}^{\mathrm{3}} +\left({ax}+{by}\right){xy} \\ $$$$\Rightarrow\mathrm{217}\left({x}+{y}\right)=\mathrm{661}+\mathrm{79}{xy}\:\:\:…\left({ii}\right) \\ $$$$\mathrm{23}×\left({ii}\right)−\mathrm{79}×\left({i}\right): \\ $$$$\left(\mathrm{23}×\mathrm{217}−\mathrm{79}×\mathrm{79}\right)\left({x}+{y}\right)=\mathrm{23}×\mathrm{661}−\mathrm{79}×\mathrm{217} \\ $$$$\Rightarrow{x}+{y}=\frac{\mathrm{194}}{\mathrm{125}} \\ $$$$\Rightarrow{xy}=\frac{\mathrm{1}}{\mathrm{23}}×\left(\mathrm{79}×\frac{\mathrm{194}}{\mathrm{125}}−\mathrm{217}\right)=−\frac{\mathrm{513}}{\mathrm{125}} \\ $$$$\left({ax}^{\mathrm{3}} +{by}^{\mathrm{3}} \right)\left({x}+{y}\right)={ax}^{\mathrm{4}} +{by}^{\mathrm{4}} +\left({ax}^{\mathrm{2}} +{by}^{\mathrm{2}} \right){xy} \\ $$$$\Rightarrow{ax}^{\mathrm{4}} +{by}^{\mathrm{4}} =\mathrm{661}×\frac{\mathrm{194}}{\mathrm{125}}+\mathrm{217}×\frac{\mathrm{513}}{\mathrm{125}}=\frac{\mathrm{47911}}{\mathrm{25}} \\ $$
Commented by Tawa11 last updated on 06/Feb/24
$$\mathrm{Nice}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 06/Feb/24
$${generally} \\ $$$${with}\:{s}_{{n}} ={ax}^{{n}} +{by}^{{n}} \\ $$$${s}_{{n}} =\left({x}+{y}\right){s}_{{n}−\mathrm{1}} −{xys}_{{n}−\mathrm{2}} \\ $$$${s}_{\mathrm{4}} =\frac{\mathrm{194}}{\mathrm{125}}×\mathrm{661}+\frac{\mathrm{513}}{\mathrm{125}}×\mathrm{217}=\frac{\mathrm{47911}}{\mathrm{25}} \\ $$$${s}_{\mathrm{5}} =\frac{\mathrm{194}}{\mathrm{125}}×\frac{\mathrm{47911}}{\mathrm{25}}+\frac{\mathrm{513}}{\mathrm{125}}×\mathrm{661}=\frac{\mathrm{17772059}}{\mathrm{3125}} \\ $$$$…… \\ $$
Commented by hardmath last updated on 06/Feb/24
$$\mathrm{perfect}\:\mathrm{solution}\:\mathrm{dear}\:\mathrm{prafessor}\:\mathrm{thank}\:\mathrm{you} \\ $$