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Topic: Equation with x^4 + x^3 + x^2 + x + 1.
Replies: 17   Last Post: Aug 10, 2012 2:55 AM

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mina_world

Posts: 2,142
Registered: 12/13/04
Equation with x^4 + x^3 + x^2 + x + 1.
Posted: Aug 7, 2012 11:36 AM
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Hello teacher~

Let c be a root of x^4 + x^3 + x^2 + x + 1.

Sequence {a_n}

a_n = [(-1)/{1+c+(c^2)+(c^3)}]^n

Find the sum{k=1 to 100} a_k.

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Sol)

c^4 + c^3 + c^2 + c = -1

==> c^3 + c^2 + c + 1 = -1/c (divide c)

==> 1/(c^3 + c^2 + c + 1) = -c

so, a_n = [(-1)/{1+c+(c^2)+(c^3)}]^n = c^n

Namely, a_n = c^n

and c^4 + c^3 + c^2 + c + 1 = 0

==> (c-1)(c^4 + c^3 + c^2 + c + 1) = 0

==> c^5 = 1

so, a_1 + a_2 + a_3 + a_4 + a_5

= c + c^2 + c^3 + c^4 + c^5

= (c + c^2 + c^3 + c^4) + c^5

= (-1) + 1 = 0

and

a_6 + a_7 + a_8 + a_9 + a_10

= c^6 + c^7 + c^8 + c^9 + c^10

= c^5(c + c^2 + c^3 + c^4 + c^5)

= 0

etc...

Thus sum{k=1 to 100} a_k = 0

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Some people(?) say that this problem has a error.

How do you think about it ?



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