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Question Number 204129 by hardmath last updated on 06/Feb/24
a , b , x , y ∈ R a + b = 23 ax + by = 79 ax^2 + by^2 = 217 ax^3 + by^3 = 661 Find: ax^4 + by^4 = ?
$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{x}\:,\:\mathrm{y}\:\in\:\mathbb{R} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{23} \\ $$$$\mathrm{ax}\:+\:\mathrm{by}\:=\:\mathrm{79} \\ $$$$\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{by}^{\mathrm{2}} \:=\:\mathrm{217} \\ $$$$\mathrm{ax}^{\mathrm{3}} \:+\:\mathrm{by}^{\mathrm{3}} \:=\:\mathrm{661} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{ax}^{\mathrm{4}} \:+\:\mathrm{by}^{\mathrm{4}} \:=\:? \\ $$
Answered by AST last updated on 06/Feb/24
x(ax+by)=79x⇒ax^2 +bxy=79x axy+by^2 =79y⇒ax^2 +by^2 +xy(a+b)=79(x+y) ⇒217=79(x+y)−23xy...(i) ax^3 +bxy^2 =217x; ax^2 y+by^3 =217y 661+xy(by+ax)=217(x+y) ⇒661=217(x+y)−79xy...(ii) ax^4 +by^4 =x(ax^3 +by^3 )+y(ax^3 +by^3 )−xy(by^2 +ax^2 ) =661x+661y−217xy 23(ii)−79(i)⇒−1250(x+y)=−1940⇒x+y=((194)/(125)) 79(ii)−217(i)⇒5130=−1250xy⇒xy=−((513)/(125)) ⇒ax^4 +by^4 =661×((194)/(125))+((217×513)/(125))=((47911)/(25))
$${x}\left({ax}+{by}\right)=\mathrm{79}{x}\Rightarrow{ax}^{\mathrm{2}} +{bxy}=\mathrm{79}{x} \\ $$$${axy}+{by}^{\mathrm{2}} =\mathrm{79}{y}\Rightarrow{ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +{xy}\left({a}+{b}\right)=\mathrm{79}\left({x}+{y}\right) \\ $$$$\Rightarrow\mathrm{217}=\mathrm{79}\left({x}+{y}\right)−\mathrm{23}{xy}…\left({i}\right) \\ $$$${ax}^{\mathrm{3}} +{bxy}^{\mathrm{2}} =\mathrm{217}{x};\:{ax}^{\mathrm{2}} {y}+{by}^{\mathrm{3}} =\mathrm{217}{y} \\ $$$$\mathrm{661}+{xy}\left({by}+{ax}\right)=\mathrm{217}\left({x}+{y}\right) \\ $$$$\Rightarrow\mathrm{661}=\mathrm{217}\left({x}+{y}\right)−\mathrm{79}{xy}…\left({ii}\right) \\ $$$${ax}^{\mathrm{4}} +{by}^{\mathrm{4}} ={x}\left({ax}^{\mathrm{3}} +{by}^{\mathrm{3}} \right)+{y}\left({ax}^{\mathrm{3}} +{by}^{\mathrm{3}} \right)−{xy}\left({by}^{\mathrm{2}} +{ax}^{\mathrm{2}} \right) \\ $$$$=\mathrm{661}{x}+\mathrm{661}{y}−\mathrm{217}{xy} \\ $$$$\mathrm{23}\left({ii}\right)−\mathrm{79}\left({i}\right)\Rightarrow−\mathrm{1250}\left({x}+{y}\right)=−\mathrm{1940}\Rightarrow{x}+{y}=\frac{\mathrm{194}}{\mathrm{125}} \\ $$$$\mathrm{79}\left({ii}\right)−\mathrm{217}\left({i}\right)\Rightarrow\mathrm{5130}=−\mathrm{1250}{xy}\Rightarrow{xy}=−\frac{\mathrm{513}}{\mathrm{125}} \\ $$$$\Rightarrow{ax}^{\mathrm{4}} +{by}^{\mathrm{4}} =\mathrm{661}×\frac{\mathrm{194}}{\mathrm{125}}+\frac{\mathrm{217}×\mathrm{513}}{\mathrm{125}}=\frac{\mathrm{47911}}{\mathrm{25}} \\ $$
Answered by mr W last updated on 06/Feb/24
(ax+by)(x+y)=ax^2 +by^2 +(a+b)xy ⇒79(x+y)=217+23xy ...(i) (ax^2 +by^2 )(x+y)=ax^3 +by^3 +(ax+by)xy ⇒217(x+y)=661+79xy ...(ii) 23×(ii)−79×(i): (23×217−79×79)(x+y)=23×661−79×217 ⇒x+y=((194)/(125)) ⇒xy=(1/(23))×(79×((194)/(125))−217)=−((513)/(125)) (ax^3 +by^3 )(x+y)=ax^4 +by^4 +(ax^2 +by^2 )xy ⇒ax^4 +by^4 =661×((194)/(125))+217×((513)/(125))=((47911)/(25))
$$\left({ax}+{by}\right)\left({x}+{y}\right)={ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +\left({a}+{b}\right){xy} \\ $$$$\Rightarrow\mathrm{79}\left({x}+{y}\right)=\mathrm{217}+\mathrm{23}{xy}\:\:\:…\left({i}\right) \\ $$$$\left({ax}^{\mathrm{2}} +{by}^{\mathrm{2}} \right)\left({x}+{y}\right)={ax}^{\mathrm{3}} +{by}^{\mathrm{3}} +\left({ax}+{by}\right){xy} \\ $$$$\Rightarrow\mathrm{217}\left({x}+{y}\right)=\mathrm{661}+\mathrm{79}{xy}\:\:\:…\left({ii}\right) \\ $$$$\mathrm{23}×\left({ii}\right)−\mathrm{79}×\left({i}\right): \\ $$$$\left(\mathrm{23}×\mathrm{217}−\mathrm{79}×\mathrm{79}\right)\left({x}+{y}\right)=\mathrm{23}×\mathrm{661}−\mathrm{79}×\mathrm{217} \\ $$$$\Rightarrow{x}+{y}=\frac{\mathrm{194}}{\mathrm{125}} \\ $$$$\Rightarrow{xy}=\frac{\mathrm{1}}{\mathrm{23}}×\left(\mathrm{79}×\frac{\mathrm{194}}{\mathrm{125}}−\mathrm{217}\right)=−\frac{\mathrm{513}}{\mathrm{125}} \\ $$$$\left({ax}^{\mathrm{3}} +{by}^{\mathrm{3}} \right)\left({x}+{y}\right)={ax}^{\mathrm{4}} +{by}^{\mathrm{4}} +\left({ax}^{\mathrm{2}} +{by}^{\mathrm{2}} \right){xy} \\ $$$$\Rightarrow{ax}^{\mathrm{4}} +{by}^{\mathrm{4}} =\mathrm{661}×\frac{\mathrm{194}}{\mathrm{125}}+\mathrm{217}×\frac{\mathrm{513}}{\mathrm{125}}=\frac{\mathrm{47911}}{\mathrm{25}} \\ $$
Commented by Tawa11 last updated on 06/Feb/24
Nice sir
$$\mathrm{Nice}\:\mathrm{sir} \\ $$
Commented by mr W last updated on 06/Feb/24
generally with s_n =ax^n +by^n s_n =(x+y)s_(n−1) −xys_(n−2) s_4 =((194)/(125))×661+((513)/(125))×217=((47911)/(25)) s_5 =((194)/(125))×((47911)/(25))+((513)/(125))×661=((17772059)/(3125)) ......
$${generally} \\ $$$${with}\:{s}_{{n}} ={ax}^{{n}} +{by}^{{n}} \\ $$$${s}_{{n}} =\left({x}+{y}\right){s}_{{n}−\mathrm{1}} −{xys}_{{n}−\mathrm{2}} \\ $$$${s}_{\mathrm{4}} =\frac{\mathrm{194}}{\mathrm{125}}×\mathrm{661}+\frac{\mathrm{513}}{\mathrm{125}}×\mathrm{217}=\frac{\mathrm{47911}}{\mathrm{25}} \\ $$$${s}_{\mathrm{5}} =\frac{\mathrm{194}}{\mathrm{125}}×\frac{\mathrm{47911}}{\mathrm{25}}+\frac{\mathrm{513}}{\mathrm{125}}×\mathrm{661}=\frac{\mathrm{17772059}}{\mathrm{3125}} \\ $$$$…… \\ $$
Commented by hardmath last updated on 06/Feb/24
perfect solution dear prafessor thank you
$$\mathrm{perfect}\:\mathrm{solution}\:\mathrm{dear}\:\mathrm{prafessor}\:\mathrm{thank}\:\mathrm{you} \\ $$

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