SOLUTION: [3 + 2√(2)]ⁿ + [3 - 2√(2)]ⁿ = 34

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Question 1155666: [3 + 2√(2)]ⁿ + [3 - 2√(2)]ⁿ = 34
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

Notice that  =  = 9 - 8 = 1.


So, if we introduce new variables  x =   and  y = , then we have this system of 2 equations in 2 unknowns


    x + y = 34

    xy    = 1


It leads to the equation in one unknown

    x +  = 34

     = 0

     =  = 


So,   =  = 33.97056,

      =  = 0.029437.


Therefore,  if x = 33.97056, then   = 33.97056;  hence  n =  = 2.


            if x = 0.029434, then   = 0.029434;  hence  n =  = -2.


CHECK.  I will check for n = 2 only.


         +  = 3^2 + 8 + 3^2 + 8 = 18 + 16 = 34.   ! Precisely correct !


ANSWER.  n = 2 and/or -2.



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