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Math Help - Mathematical Induction Question

  1. #1
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    Mathematical Induction Question

    I was hoping someone could help me with a proof. I have attached an image.
    I am going to prove it using induction (rather than well-ordering), and the base cases where n=1,2 and 3 are straightforward.

    Thank you in advance.
    Attached Thumbnails Attached Thumbnails Mathematical Induction Question-1.jpg  
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  2. #2
    Jen
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    Inductive Hypothesis:

    \left(\frac{5}{4}\right)^k-\left(\frac{3}{4}\right)^k>\frac{k}{2}

    Show:

    \left(\frac{5}{4}\right)^{k+1}-\left(\frac{3}{4}\right)^{k+1}>\frac{k+1}{2}

    Working with the left side...

    \left(\frac{5}{4}\right)\left(\frac{5}{4}\right)^k-\left(\frac{3}{4}\right)\left(\frac{3}{4}\right)^k  >\left(\frac{5}{4}\right)\left(\frac{5}{4}\right)^  k-\left(\frac{5}{4}\right)\left(\frac{3}{4}\right)^k

    \left(\frac{5}{4}\right)\left(\frac{5}{4}\right)^k-\left(\frac{3}{4}\right)\left(\frac{3}{4}\right)^k  >\frac{5}{4}\left[\left(\frac{5}{4}\right)^k-\left(\frac{3}{4}\right)^k\right]

    On the right hand side we have 5/4 times our inductive hypothesis, substituting we get...

    \left(\frac{5}{4}\right)\left(\frac{5}{4}\right)^k-\left(\frac{3}{4}\right)\left(\frac{3}{4}\right)^k  >\frac{5}{4}\left(\frac{k}{2}\right)=\left(1+\frac  {1}{4}\right)\left(\frac{k}{2}\right)=\frac{k}{2}+  \frac{k}{8}>\frac{k}{2}+\frac{1}{2}

    \frac{k}{8}>\frac{1}{2} Because k>4

    So, by transitivity, we get...

    \left(\frac{5}{4}\right)\left(\frac{5}{4}\right)^k-\left(\frac{3}{4}\right)\left(\frac{3}{4}\right)^k  >\frac{k}{2}+\frac{1}{2}

    W^5 "Which Was What We Wanted"

    Sorry I didn't know how to do the greater than or equal to symbol.

    Hope that was clear enough.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Jen View Post
    Sorry I didn't know how to do the greater than or equal to symbol.
    just type \ge

    type \le for less than or equal to

    type \ne for not equal to
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  4. #4
    Jen
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    Oooops, lets try this again.
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  5. #5
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    Thank you very much Jen. Now I just need to do c using part b as my solution.
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  6. #6
    Jen
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    Given that:

    \left(\frac{5}{4}\right)-\left(\frac{3}{4}\right)^x>\left(\frac{x}{2}\right  )

    Show:

    3^x+4^x<5^x

    Using the given inequality... (multiply both sides by 4^x, note that this is legal because exponential functions are non-negative)

    4^x\left[\left(\frac{5}{4}\right)-\left(\frac{3}{4}\right)^x>\left(\frac{x}{2}\right  )\right]

    5^x-3^x>4^x\frac{x}{2}

    Adding 3^x to both sides we get...

    5^x>4^x\frac{x}{2}+3^x>4^x+3^x

    The inequality on the right is possible because x is always greater than or equal to 2 so x/2 is always greater than or equal to 1.

    Once again by transitivity we get

    5^x>4^x+3^x



    Once again, hope that made sense...
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