A positive integer n is randomly guesses. Let the probability that (xⁿ⁺¹- xⁿ +1) is divisible by (x² - x +1) is P . Find P

This question is based on complex numbers so.....

 

Let’s assume an equation  

 

therefore,

 

So all roots of the above cubic equation except 1 are roots of  

 

Now we know roots of are  

  

So      ,       ,    

 

Clearly roots of the first equation are     ,     ,

 

putting  in and simplifying we get, [we can put , but i’ll come to that later][step a]

 

                                                      [ ]

 

 

Since a power of    is giving a real number, that power must be a multiple of 3.  Hence n+2 is divisible by 3

 

also the only real result of a power of omega is 1. hence the power of -1 is even. Hence n+3 is divisible by 2

 

 

let’s now list out the possibilties,..

 

Notice how every 6 possibilities there is one such number

 

let the sample space thus be 6*q [q being any integer], there would be q favourable outcomes.

as the sample space is infinite (no restriction given) we can assume that even though the sample space is infinite this ‘infinity’ is divisible by 6 [any other case would be mathematically insignificant]

 

thus probability is 

 

the same will actually happen if in [step a] instead of  we put 

Since a power of    is giving a real number, that power must be a multiple of 3.  Hence 2n+1 is divisible by 3

 

also the only real result of a power of omega is 1. hence the power of -1 is even. Hence n+3 is divisible by 2

 

 

listing out the possibilities again....

 

the same result the same equation and thus the same value:

 

hence,

 

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