- #1

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Without actually solving for x, prove the following: if x + 1/x = 1, then x

^{7}+ 1/x

^{7}= 1.

Assume that x

Then:

x

x

x

Therefore x is an imaginary number.

Now by hypothesis we know that x + x

Since x is imaginary, x

By the closure property of the set of imaginary numbers, x + x

However, the hypothesis states that x + x

^{7}+ 1/x^{7}= 0.Then:

x

^{7}= -1/x^{7}x

^{7}x^{7}= -1x

^{14}= -1.Therefore x is an imaginary number.

Now by hypothesis we know that x + x

^{-1}= 1.Since x is imaginary, x

^{-1}is also imaginary.By the closure property of the set of imaginary numbers, x + x

^{-1}must also be imaginary.However, the hypothesis states that x + x

^{-1}= 1, which is not imaginary. Thus by contradiction, if x + 1/x = 1, then x^{7}+ 1/x^{7}= 1.Does this work at all? Not only am I not sure, I feel like I cheated by just stating that x

^{7}+ 1/x

^{7}= 0. What I was attempting to do was assume ~Q and then derive a contradiction using P, using the basic proof by contradiction technique. Saying it is equal to 0 is more specific than saying it is not equal to 1, but either way it's still stating it isn't equal to 1. Is this even a viable way to approach this proof?

Thanks!

EDIT- I should point out that the book actually does a direct proof using just a bit of recursive algebraic manipulation. I just want to see if this way is legal.