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Question Number 205545 by Abdullahrussell last updated on 24/Mar/24

Commented by mr W last updated on 24/Mar/24

$$=768612?$$

Answered by A5T last updated on 24/Mar/24

$$x+\frac{1}{x}=a;y+\frac{1}{y}=b\Rightarrow a+b=12$$$${(x+\frac{1}{x})}^2+{(y+\frac{1}{y})}^2=74\Rightarrow a^2+b^2=74$$$$\Rightarrow {(a+b)}^2-2ab=74\Rightarrow 2ab=70\Rightarrow ab=35$$$$\Rightarrow z^2-12z+35=0\Rightarrow a=7orb=5uptosymmetry$$$$\Rightarrow x+\frac{1}{x}=7\Rightarrow x^2-7x+1=0;y+\frac{1}{y}=5\Rightarrow y^2-5y+1=0$$$$\Rightarrow x=\frac{7\underset{-}{+}3\sqrt{5}}{2};y=\frac{5\underset{-}{+}\sqrt{21}}{2}$$$$\Rightarrow x^7+\frac{1}{x^7}+\frac{1}{y^7}+y^7=768612$$

Answered by Rasheed.Sindhi last updated on 24/Mar/24

$$x+\frac{1}{x}=a,y+\frac{1}{y}=12-a$$$$x^2+\frac{1}{x^2}+2+y^2+\frac{1}{y^2}+2=70+4=74$$$${(x+\frac{1}{x})}^2+{(y+\frac{1}{y})}^2=74$$$$a^2+{(12-a)}^2=74$$$$a^2+144-24a+a^2-74=0$$$$2a^2-24a+70=0$$$$a^2-12a+35=0$$$$(a-5)(a-7)=0$$$$a=5,7$$$$x+\frac{1}{x}=5,7$$$$y+\frac{1}{y}=7,5$$$$\bullet x+\frac{1}{x}=5\mathrm{\&}y+\frac{1}{y}=7$$$$[obviously,x+\frac{1}{x}=7\mathrm{\&}y+\frac{1}{y}=5willgivesameresult]$$$$\bullet x^2=5x-1\mathrm{\&}{y}^2=7x-1$$$$\bullet x^3=5x^2-x=5(5x-1)-x=24x-5$$$$\bullet x^4=24x^2-5x$$$$=24(5x-1)-5x=115x-24$$$$\bullet x^7=x^3.x^4=(24x-5)(115x-24)$$$$=2760x^2-1151x+120$$$$=2760(5x-1)-1151x+120$$$$=12649x-2640$$$$$$$$\bullet \frac{1}{x}=5-x\Rightarrow \bullet \frac{1}{x^2}=\frac{5}{x}-1=5(5-x)-1$$$$=24-5x$$$$\bullet \frac{1}{x^3}=24\left(\frac{1}{x}\right)-5=24(5-x)-5$$$$=115-24x$$$$\bullet \frac{1}{x^4}=115\left(\frac{1}{x}\right)-24=115(5-x)-24$$$$=551-115x$$$$\bullet \frac{1}{x^7}=\frac{1}{x^3}\cdot \frac{1}{x^4}=(115-24x)(551-115x)$$$$=2760x^2-26449x+63365$$$$=2760(5x-1)-26449x+63365$$$$=60605-12649x$$$$\bullet x^7+\frac{1}{x^7}$$$$=(12649x-2640)+(60605-12649x)$$$$=57965$$$$\bullet y+\frac{1}{y}=7\Rightarrow \bullet y^2=7y-1$$$$\bullet y^3=7y^2-y=7(7y-1)-y=48y-7$$$$\bullet y^4=48y^2-7y=48(7y-1)-7y$$$$=329x-48$$$$\bullet y^7=y^3\cdot y^4=(48y-7)(329y-48)$$$$=15792y^2-4607y+336$$$$=15792(7y-1)-4607y+336$$$$=105937y-15456$$$$$$$$\bullet \frac{1}{y}=7-y\Rightarrow$$$$\bullet \frac{1}{y^2}=\frac{7}{y}-1=7(7-y)-1=48-7y$$$$\bullet \frac{1}{y^3}=\frac{48}{y}-7=48(7-y)-7=329-48y$$$$\bullet \frac{1}{y^4}=\frac{329}{y}-48=329(7-y)-48$$$$=2255-329y$$$$\bullet \frac{1}{y^7}=\frac{1}{y^3}\cdot \frac{1}{y^4}=(329-48y)(2255-329y)$$$$=15792y^2-216481y+741895$$$$=15792(7y-1)-216481y+741895$$$$=726103-105937y$$$$\bullet y^7+\frac{1}{y^7}=(105937y-15456)+(726103-105937y)$$$$=710647$$$$\bullet x^7+\frac{1}{x^7}+y^7+\frac{1}{y^7}=57965+710647$$$$=768612$$

Answered by mr W last updated on 24/Mar/24

$$x+\frac{1}{x}=a,say$$$$\Rightarrow x^2+\frac{1}{x^2}=a^2-2$$$$(x^2+\frac{1}{x^2})(x+\frac{1}{x})=a^3-2a$$$$x^3+\frac{1}{x^3}+x+\frac{1}{x}=a^3-2a$$$$\Rightarrow x^3+\frac{1}{x^3}=a^3-3a$$$$(x^3+\frac{1}{x^3})(x^2+\frac{1}{x^2})=(a^3-3a)(a^2-2)=a^5-5a^3+6a$$$$x^5+\frac{1}{x^5}+x+\frac{1}{x}=a^5-5a^3+6a$$$$\Rightarrow x^5+\frac{1}{x^5}=a^5-5a^3+5a$$$$(x^5+\frac{1}{x^5})(x^2+\frac{1}{x^2})=(a^5-5a^3+5a)(a^2-2)=a^7-7a^5+15a^3-10a$$$$x^7+\frac{1}{x^7}+x^3+\frac{1}{x^3}=(a^5-5a^3+5a)(a^2-2)=a^7-7a^5+15a^3-10a$$$$\Rightarrow x^7+\frac{1}{x^7}=a^7-7a^5+14a^3-7a$$$$similarly$$$$y+\frac{1}{y}=b$$$$y^7+\frac{1}{y^7}=b^7-7b^5+14b^3-7b$$$$S=x^7+\frac{1}{x^7}+y^7+\frac{1}{y^7}=(a^7+b^7)-7(a^5+b^5)+14(a^3+b^3)-7(a+b)$$$$given:$$$$x+\frac{1}{x}+y+\frac{1}{y}=12\Rightarrow a+b=12$$$$x^2+\frac{1}{x^2}+y^2+\frac{1}{y^2}=70\Rightarrow a^2+b^2=74$$$${(a+b)}^2-(a^2+b^2)={12}^2-74$$$$\Rightarrow ab=\frac{{12}^2-74}{2}=35$$$$(a^2+b^2)(a+b)=74\times 12$$$$a^3+b^3+ab(a+b)=74\times 12$$$$\Rightarrow a^3+b^3=74\times 12-35\times 12=468$$$$(a^3+b^3)(a^2+b^2)=468\times 74$$$$a^5+b^5+{\left(ab\right)}^2(a+b)=468\times 74$$$$\Rightarrow a^5+b^5=468\times 74-{35}^2\times 12=19932$$$$(a^5+b^5)(a^2+b^2)=19932\times 74$$$$a^7+b^7+{\left(ab\right)}^2(a^3+b^3)=19932\times 74$$$$\Rightarrow a^7+b^7=19932\times 74-{35}^2\times 468=901668$$$$$$$$S=901668-7\times 19932+14\times 468-7\times 12$$$$=768612✓$$

Answered by mr W last updated on 24/Mar/24

$$<<using{newton}^{\prime }sidentities>>$$$$a_1=x+\frac{1}{x}=e_1=a,say$$$$a_2=e_1{a}_1-2e_2=a^2-2with{e}_2=1$$$$a_3=e_1{a}_2-e_2{a}_1=a(a^2-2)-a=a^3-3a$$$$a_4=e_1{a}_3-e_2{a}_2=a(a^3-3a)-(a^2-2)=a^4-4a^2+2$$$$a_5=e_1{a}_4-e_2{a}_3=a(a^4-4a^2+2)-(a^3-3a)=a^5-5a^3+5a$$$$a_6=e_1{a}_5-e_2{a}_4=a(a^5-5a^3+5a)-(a^4-4a^2+2)=a^6-6a^4+9a^2-2$$$$a_7=e_1{a}_6-e_2{a}_5=a(a^6-6a^4+9a^2-2)-(a^5-5a^3+5a)=a^7-7a^5+14a^3-7a$$$$i.e.x^7+\frac{1}{x^7}=a^7-7a^5+14a^3-7awitha=x+\frac{1}{x}$$$$similarly$$$$y^7+\frac{1}{y^7}=b^7-7b^5+14b^3-7bwithb=y+\frac{1}{y}$$$$x^7+\frac{1}{x^7}+y^7+\frac{1}{y^7}=(a^7+b^7)-7(a^5+b^5)+14(a^3+b^3)-7(a+b)$$$$$$$$p_1=a+b=x+\frac{1}{x}+y+\frac{1}{y}=12=e_1$$$$p_2=a^2+b^2={(x+\frac{1}{x})}^2+{(y+\frac{1}{y})}^2=x^2+\frac{1}{x^2}+2+y^2+\frac{1}{y^2}+2=70+4=74$$$$p_2=e_1{p}_1-2e_2=12\times 12-2e_2=74\Rightarrow e_2=35$$$$p_3=e_1{p}_2-e_2{p}_1=12\times 74-35\times 12=468$$$$p_4=e_1{p}_3-e_2{p}_2=12\times 468-35\times 74=3026$$$$p_5=e_1{p}_4-e_2{p}_3=12\times 3026-35\times 468=19932$$$$p_6=e_1{p}_5-e_2{p}_4=12\times 19932-35\times 3026=133274$$$$p_7=e_1{p}_6-e_2{p}_5=12\times 133274-35\times 19932=901668$$$$$$$$x^7+\frac{1}{x^7}+y^7+\frac{1}{y^7}=901668-7\times 19932+14\times 468-7\times 12$$$$=768612✓$$

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