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Question Number 217730 by ArshadS last updated on 19/Mar/25
x+(1/x)=3 ; x^5 +(1/x^5 )=?
x+1x=3;x5+1x5=?
Answered by Wuji last updated on 19/Mar/25
let S_n =x^n +(1/x^n ) S_1 =x+(1/x)=3 S_2 =(x+(1/x))^2 −2 ⇒S_2 =3^2 −2=7 S_3 =(x+(1/x))S_2 −S_1 ⇒S_3 =3×7−3=18 S_4 =(x+(1/x))S_3 −S_2 ⇒S_4 =3×18−7=47 S_5 =(x+(1/x))S_4 −S_3 ⇒S_5 =3×47−18=123 S_5 =x^5 +(1/x^5 )=123
letSn=xn+1xnS1=x+1x=3S2=(x+1x)2−2⇒S2=32−2=7S3=(x+1x)S2−S1⇒S3=3×7−3=18S4=(x+1x)S3−S2⇒S4=3×18−7=47S5=(x+1x)S4−S3⇒S5=3×47−18=123S5=x5+1x5=123
Commented by ArshadS last updated on 22/Mar/25
Thanks sir!
Thankssir!
Answered by Rasheed.Sindhi last updated on 19/Mar/25
x+(1/x)=3 ; x^5 +(1/x^5 )=? x+(1/x)=3⇒x^2 =3x−1 ⇒x^4 =9x^2 −6x+1 =9(3x−1)−6x+1 =21x−8 ⇒ determinant ((x^5 ))=21x^2 −8x=21(3x−1)−8x =55x−21 (1/x)=3−x (1/x^2 )=(3−x)^2 =x^2 −6x+9 =(3x−1)−6x+9=−3x+8 (1/x^4 )=(−3x+8)^2 =9x^2 −48x+64 =9(3x−1)−48x+64 =27x−9−48x+64 =−21x+55 determinant (((1/x^5 )))=−21+55((1/x))=−21+55(3−x) =−21+165−55x=−55x+144 determinant (((x^5 +(1/x^5 ))))=(55x−21)+(−55x+144) =123
x+1x=3;x5+1x5=?x+1x=3⇒x2=3x−1⇒x4=9x2−6x+1=9(3x−1)−6x+1=21x−8⇒x5=21x2−8x=21(3x−1)−8x=55x−211x=3−x1x2=(3−x)2=x2−6x+9=(3x−1)−6x+9=−3x+81x4=(−3x+8)2=9x2−48x+64=9(3x−1)−48x+64=27x−9−48x+64=−21x+551x5=−21+55(1x)=−21+55(3−x)=−21+165−55x=−55x+144x5+1x5=(55x−21)+(−55x+144)=123
Commented by ArshadS last updated on 22/Mar/25
Thanks sir!
Thankssir!
Answered by Ghisom last updated on 20/Mar/25
x^2 −3x+1=0 x^2 =3x−1 x^5 =x(x^2 )^2 =x(3x−1)^2 =x(9x^2 −6x+1)= =x(9(3x−1)−6x+1)=x(21x−8)= =21x^2 −8x=21(3x−1)−8x=55x−21 (1/x^5 )=(1/(55x−21))=(1/(55(3−(1/x))−21))=(x/(144x−55)) 55x−21+(x/(144x−55))= =(((55x−21)(144x−55)+x)/(144x−55))= =((7920x^2 −6048x+1155)/(144x−55))= =((7920(3x−1)−6048x+1155)/(144x−55))= =((17712x−6765)/(144x−55))=123
x2−3x+1=0x2=3x−1x5=x(x2)2=x(3x−1)2=x(9x2−6x+1)==x(9(3x−1)−6x+1)=x(21x−8)==21x2−8x=21(3x−1)−8x=55x−211x5=155x−21=155(3−1x)−21=x144x−5555x−21+x144x−55==(55x−21)(144x−55)+x144x−55==7920x2−6048x+1155144x−55==7920(3x−1)−6048x+1155144x−55==17712x−6765144x−55=123
Commented by ArshadS last updated on 22/Mar/25
Thanks sir!
Thankssir!

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