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Math Help - -1^n = 1 or -1

  1. #1
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    (-1)^n = 1 or -1

    Okay, this might be a little bit stupid question...
    I'm studying ross elementary analysis.
    And I'm trying to get some skills in writing formal proofs, instead of intuitive ideas.

    I want to proof that the sequence (-1)^n = 1 or -1.
    But how do I give a real proof for this ?
    Last edited by kasper90; September 5th 2012 at 08:15 AM.
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  2. #2
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    Re: -1^n = 1 or -1

    Quote Originally Posted by kasper90 View Post
    Okay, this might be a little bit stupid question...
    I'm studying ross elementary analysis.
    And I'm trying to get some skills in writing formal proofs, instead of intuitive ideas.

    I want to proof that the sequence -1^n = 1 or -1.
    But how do I give a real proof for this ?
    -1^n = -1, since you need to do the exponentiation before the negation.
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  3. #3
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    Re: (-1)^n = 1 or -1

    I actually meant to say (-1)^n.
    sorry for that
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    Re: -1^n = 1 or -1

    (-1)^n = 1 if n is even, and (-1)^n = -1 if n is odd.
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  5. #5
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    Re: -1^n = 1 or -1

    Okay, but how do I prove that ?
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    Re: -1^n = 1 or -1

    Quote Originally Posted by kasper90 View Post
    Okay, but how do I prove that ?
    If n is even, n=2k then (-1)^n=(-1)^{2k}=[(-1)^{2}]^k=[1]^k=1.

    If n is odd, n=2k+1 then (WHAT?)
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  7. #7
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    Re: -1^n = 1 or -1

    Okay, I'll try:
    If n is even,
    than ∃k∈N : n = 2k,
    than (-1)^n = (-1)^2k=[(-1)^2]^k = 1^k

    If n is uneven,
    than ∃k∈N : n=2k-1,
    than (-1)^n = (-1)^[2k-1] = -1^k,

    Induction (prove that 1^k =1)
    1^1
    if 1^k = 1, than 1^(k+1)=1^k * 1 = 1^k = 1
    By induction, 1^k =1 for all natural numbers.
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  8. #8
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    Re: -1^n = 1 or -1

    Plato posted his post just before I did,
    but I think I got the idea, right ?

    Is this a valid proof ?
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    Re: -1^n = 1 or -1

    Quote Originally Posted by kasper90 View Post
    Plato posted his post just before I did,
    but I think I got the idea, right ? Is this a valid proof ?
    You have proved that (-1)^{2k}=1 so use it.
    (-1)^{2k+1}=(-1)^{2k}(-1)^{1}=(1)(-1)=-1.
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