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Math Help - Principle of Induction Proof

  1. #1
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    Principle of Induction Proof

    Prove that 7^(2n-1) + 13^(2n-1) is divisible by 10 for all n in N. I know this is done using algebra and followed by the definition of divisibility. I just got a bit lost in the algebra.
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  2. #2
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    The induction step:
    Assume that 10 divides 7^{2n-1}+13^{2n-1}.
    Want to show for n+1, 1.e want to show that 10 divides
    7^{2(n+1)-1}+13^{2(n+1)-1}=7^{2n+1}+13^{2n+1}=49.7^{2n-1}+169.13^{2n-1}

    Let see if you can go from here............
    Last edited by watchmath; October 10th 2008 at 08:44 AM.
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  3. #3
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    shouldn't that n-1 you inserted be n+1 ??
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  4. #4
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    Quote Originally Posted by Caity View Post
    shouldn't that n-1 you inserted be n+1 ??
    Yes I have fixed it now.
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  5. #5
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    Also the 49 and 169 be 7 and 13 since they were factored out...??? Sorry I get confused easily..
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  6. #6
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    never mind you factored out 2 of each of them I see...
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  7. #7
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    Thanks I think I got it... I let the original eqation equal 10j j is in N. and ended up with 10(40j + 12 times 13^(2n-1)) which is in N so original is divisible by N. Thanks again...
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  8. #8
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    Here's another one and not sure what to do with the exponents on this... Prove the if 0<a<b then a^(1/n) < b^(1/n) for n in N. Or how to show for n + 1.

    Oh nm this would be nth roots so prove by contradiction... sorry I answered my own question..
    Last edited by Caity; October 10th 2008 at 10:48 AM. Reason: Realizaion
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  9. #9
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    To make it work the statement should be for every 0<a<b any any n in N we have a^(1/n)<b^(1/n).

    Now we want to show a^(1/(n+1))<b^(1/(n+1)). Can you find A and B such that a^(1/(n+1))=A^(1/n) and b^(1/(n+1))=B^(1/n) ?
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  10. #10
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    Quote Originally Posted by watchmath View Post
    To make it work the statement should be for every 0<a<b any any n in N we have a^(1/n)<b^(1/n).

    Now we want to show a^(1/(n+1))<b^(1/(n+1)). Can you find A and B such that a^(1/(n+1))=A^(1/n) and b^(1/(n+1))=B^(1/n) ?
    I know that is what I need to do. How does the algebra go on this? Do you think nth root contradiction proof would maybe be easier to show this?
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