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If x^2+x+1=0 then find the value of (X+1/x)^2+ (x^2+1/x^2)+(x^3+1/x^3)^2+................(x^27+1/x^27) I have attached the image of the question. (HINT: maybe it`s related to come Roots of unity)

If x^2+x+1=0 then find the value of (X+1/x)^2+ (x^2+1/x^2)+(x^3+1/x^3)^2+................(x^27+1/x^27) I have attached the image of the question. (HINT: maybe it`s related to come Roots of unity)

Question Image
Grade:11

1 Answers

Arun
25763 Points
3 years ago
Dear Piyush
 
roots are  w and w2
Remember
1+ w + w2 = 0
w3 = 1
1/w = w2    & 1/w2 = w
hence
(w + 1/w)2 + (w2 + 1/w2)2 + (w3 + 1/w3)2 ….....
now 
(w + w^2)^2 + (w^2 + w)^2 + 2^2 + (w + w^2)^2 + (w^2 + w)^2 + 2^2 + …....
 = ( – 1)2 + ( –1)2 + 4 + ( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4+( – 1)2 + ( –1)2 + 4
 = 36 + 18
 = 54
 
Regards
Arun (askIITians forum expert)

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