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Re: Equation with x^4 + x^3 + x^2 + x + 1.
Posted:
Aug 7, 2012 1:23 PM
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2012? 8? 8? ??? ?? 2? 13? 23? UTC+9, dilettante ?? ?:
> "Mina" <mina_world@hanmail.net> wrote in message news:91f9f750-14b3-416f-b264-15a3a86a1bd8@googlegroups.com... > 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. > > --------------------------------------------- > 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 > > ------------------------------------------------- > Some people(?) say that this problem has a error. > > How do you think about it ? Perhaps the error is supposed to be that the roots of the equation aren't real,but I don't see that as an error. When it says let "c be a root of " and then gives an equation that has only non-real roots, the natural assumption is that we are working with complex numbers.
Hm.. If c is a complex number,
the answer with sum{k=1 to 100} a_k vary ?
why?
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