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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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