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Question Number 204129 by hardmath last updated on 06/Feb/24

$$\mathrm{a},\mathrm{b},\mathrm{x},\mathrm{y}\in \mathbb{R}$$$$\mathrm{a}+\mathrm{b}=23$$$$\mathrm{ax}+\mathrm{by}=79$$$${\mathrm{ax}}^2+{\mathrm{by}}^2=217$$$${\mathrm{ax}}^3+{\mathrm{by}}^3=661$$$$\mathrm{Find}:{\mathrm{ax}}^4+{\mathrm{by}}^4=?$$

Answered by deleteduser1 last updated on 06/Feb/24

$$x(ax+by)=79x\Rightarrow {ax}^2+bxy=79x$$$$axy+{by}^2=79y\Rightarrow {ax}^2+{by}^2+xy(a+b)=79(x+y)$$$$\Rightarrow 217=79(x+y)-23xy...\left(i\right)$$$${ax}^3+{bxy}^2=217x;{ax}^{2}y+{by}^3=217y$$$$661+xy(by+ax)=217(x+y)$$$$\Rightarrow 661=217(x+y)-79xy...\left(ii\right)$$$${ax}^4+{by}^4=x({ax}^3+{by}^3)+y({ax}^3+{by}^3)-xy({by}^2+{ax}^2)$$$$=661x+661y-217xy$$$$23\left(ii\right)-79\left(i\right)\Rightarrow -1250(x+y)=-1940\Rightarrow x+y=\frac{194}{125}$$$$79\left(ii\right)-217\left(i\right)\Rightarrow 5130=-1250xy\Rightarrow xy=-\frac{513}{125}$$$$\Rightarrow {ax}^4+{by}^4=661\times \frac{194}{125}+\frac{217\times 513}{125}=\frac{47911}{25}$$

Answered by mr W last updated on 06/Feb/24

$$(ax+by)(x+y)={ax}^2+{by}^2+(a+b)xy$$$$\Rightarrow 79(x+y)=217+23xy...\left(i\right)$$$$({ax}^2+{by}^2)(x+y)={ax}^3+{by}^3+(ax+by)xy$$$$\Rightarrow 217(x+y)=661+79xy...\left(ii\right)$$$$23\times \left(ii\right)-79\times \left(i\right):$$$$(23\times 217-79\times 79)(x+y)=23\times 661-79\times 217$$$$\Rightarrow x+y=\frac{194}{125}$$$$\Rightarrow xy=\frac{1}{23}\times (79\times \frac{194}{125}-217)=-\frac{513}{125}$$$$({ax}^3+{by}^3)(x+y)={ax}^4+{by}^4+({ax}^2+{by}^2)xy$$$$\Rightarrow {ax}^4+{by}^4=661\times \frac{194}{125}+217\times \frac{513}{125}=\frac{47911}{25}$$

Commented by Tawa11 last updated on 06/Feb/24

$$\mathrm{Nice}\mathrm{sir}$$

Commented by mr W last updated on 06/Feb/24

$$generally$$$$with{s}_n={ax}^n+{by}^n$$$$s_n=(x+y)s_{n-1}-{xys}_{n-2}$$$$s_4=\frac{194}{125}\times 661+\frac{513}{125}\times 217=\frac{47911}{25}$$$$s_5=\frac{194}{125}\times \frac{47911}{25}+\frac{513}{125}\times 661=\frac{17772059}{3125}$$$$......$$

Commented by hardmath last updated on 06/Feb/24

$$\mathrm{perfect}\mathrm{solution}\mathrm{dear}\mathrm{prafessor}\mathrm{thank}\mathrm{you}$$

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