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Thread: Specific Problem using Induction

  1. #1
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    Question Specific Problem using Induction

    My buddy and I cannot figure out how to do this problem in preparation for an exam. It goes:

    Let A(n) be a sequence defined by A(1) = 3.
    A(n) = (2 * A(n-1))^0.5, for all integers n >= 2.
    Prove that for all A(n), A(n) > 2.

    How can I go about this?
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  2. #2
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    Re: Specific Problem using Induction

    I would look at the first several terms in the sequence:

    $\displaystyle A_1=3$

    $\displaystyle A_2=2^{\Large\frac{1}{2}}\cdot3^{\Large\frac{1}{2} }$

    $\displaystyle A_3=2^{\Large\frac{1}{2}}\left(2^{\Large\frac{1}{2 }}\cdot3^{\Large\frac{1}{2}}\right)^{\Large\frac{1 }{2}}=2^{\Large\frac{3}{4}}\cdot3^{\Large\frac{1}{ 4}}$

    I think at this point, we may state our induction hypothesis $\displaystyle P_n$:

    $\displaystyle A_n=2^{\Large\frac{2^{n-1}-1}{2^{n-1}}}\cdot3^{\Large\frac{1}{2^{n-1}}}$

    I would use the recursive definition as the induction step...what do you get?
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    Post Re: Specific Problem using Induction

    I have come to a similar conclusion, except my formula is:
    A(n) = (20.5)n-1 * ((3)0.5)n-1

    My question is more how can I use this to explain that it never goes below 2? I am lost in this part of the process.
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    MHF Contributor MarkFL's Avatar
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    Re: Specific Problem using Induction

    Quote Originally Posted by oli1230 View Post
    I have come to a similar conclusion, except my formula is:
    A(n) = (20.5)n-1 * ((3)0.5)n-1

    My question is more how can I use this to explain that it never goes below 2? I am lost in this part of the process.
    Do you see how your formula doesn't work? Have you tried to see if you can prove it by induction? I have verified that the induction hypothesis I gave can be proven. I think we should get the correct formula first.
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    Re: Specific Problem using Induction

    Of course the problem, as initially stated, doesn't require that we find a formula, just that we show that every term is greater than 2. We are given that A(1)= 3> 2 so that is taken care of. Now, suppose that, for some k, A(k)> 2. Since A(k)> 2, 2A(k)> 4 and then immediately A(k+ 1)= (2A(k))^{1/2}> 2.
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    Re: Specific Problem using Induction

    Thank you so much for your help, we have figured out the problem based on your input greatly appreciated!
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    MHF Contributor MarkFL's Avatar
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    Re: Specific Problem using Induction

    Since the thread title said to use induction, I naturally assumed that you were instructed to do so.
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  8. #8
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    Re: Specific Problem using Induction

    I used an amalgamation of your formula and his logic to put it in an inductive proof.
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