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Math Help - Induction

  1. #1
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    Induction

    Assume that x+(1/x) is an integer, how do I, by using induction, show that x^2 + (1/x^2) , x^3 + (1/x^3), .... , x^n + (1/x^n) are also integers?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello

    First, show that x^2+\frac{1}{x^2} lies in  \mathbb{N} by developping \left(x+\frac{1}{x}\right)\left(x+\frac{1}{x}\righ  t).

    Then, assuming there exists an integer n such that x^n+\frac{1}{x^n} and x^{n-1}+\frac{1}{x^{n-1}} lie in \mathbb{N}, try to develop \left(x^{n}+\frac{1}{x^{n}}\right)\left(x+\frac{1}  {x}\right) .
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  3. #3
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    So by assuming that there exists an integer such that and lie in , what we then do is to show that this is also true for n+1 ?
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Yes. I should have written it, sorry.
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  5. #5
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    would it make any difference if we assumed that it's true for n+1, and for n+2, and by using this proving that it's also true for n?
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  6. #6
    Super Member flyingsquirrel's Avatar
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    Yes, it would make a difference : it would be false.

    Let's call P(n) the relation " x^n+\frac{1}{x^n} is an integer"
    If one shows that if P(n-1) and P(n) are true then P(n+1) is true too, as P(1) and P(2) are true, we get that P(3) is true. Then, as P(2) and P(3) are true, we get that P(4) is true... so we'll reach any integer.

    Instead, if you show that P(n+1) and P(n+2) true imply P(n) true, you go decreasing and won't reach all the integers...
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