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Math Help - Proof by induction

  1. #1
    Super Member Showcase_22's Avatar
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    Proof by induction

    I have this question and I think i've done it. The trouble is, thinking i've done it isn't enough and I need to know if I actually have!

    Question: Use induction to prove that if both x and y are positive then x<y \Rightarrow x^n<y^n.

    I did this:

    I rearranged it like so \frac {x}{y} <1 \Rightarrow \frac {x^n}{y^n} <1

    For n=1:

    \frac {x^1}{y^1}<1 \Rightarrow \frac {x}{y} <1

    Therefore it is true for n=1.

    For n=k:

    \frac {x^k}{y^k} <1

    For n=k+1:

    \frac {x^{k+1}}{y^{k+1}}<1

    \frac {x^k}{y^k} \frac {x}{y}<1

    \frac {x^k}{y^k}< \frac {y}{x}

    We know that \frac{y}{x}>1 and \frac{x^k}{y^k}<1 so this inequality makes sense.

    At this point I think it has been proven true for n=k+1.

    So is that the end of the proof? I still have to write the little thing on the end explaining it was proof by induction. I just want to know if this is the right method.
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  2. #2
    MHF Contributor
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    It looks ok in general. The only nit-pick I have is the order of demonstrating n ->n+1. You kind of assumed n+1 was true then justified it after the fact. I would just reverse the order so that it flows more naturally going from n, doing some algebra, and implying n+1.

    But yes, once you show n->n+1 you are basically done. You only have to make some sentence wrapping up the proof.
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  3. #3
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    Induction seems like a bit of overkill for this problem, as you're raising a number that is less than one to a positive power. This is almost trivial.

    Bobak
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  4. #4
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    Quote Originally Posted by bobak View Post
    Induction seems like a bit of overkill for this problem, as you're raising a number that is less than one to a positive power. This is almost trivial.

    Bobak
    True. But if a question prescribes a method, then that's the method that has to be used. Especially important to remember this in exams.
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  5. #5
    Super Member Showcase_22's Avatar
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    It looks ok in general. The only nit-pick I have is the order of demonstrating n ->n+1. You kind of assumed n+1 was true then justified it after the fact. I would just reverse the order so that it flows more naturally going from n, doing some algebra, and implying n+1.
    Okay great! It's nice to know that i'm doing it correctly.

    What do you mean "reverse the order"?

    Do you mean I should write something like "assume true for n=k" or do you mean I should write \frac{x^{k+1}}{y^{k+1}} and not put the <1 part in?

    Induction seems like a bit of overkill for this problem, as you're raising a number that is less than one to a positive power. This is almost trivial.
    As Mr Fantastic said, it does say to use this method. I was also thinking it would just be easier to say that \frac {x}{y} is less than one (but greater than 0). When a fraction is raised to a positive power it would get smaller but instead I have to do it a long way.
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