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Mathmatical Induction Problem (Divisibility)

  1. May 10, 2007 #1
    1. The problem statement, all variables and given/known data
    Use Mathematical Induction to prove that [tex] 12^n + 2(5^{n-1}) [/tex] is divisible by 7 for all [tex] n \in Z^+ [/tex]

    2. Relevant equations



    3. The attempt at a solution

    First, show that it works for n = 1:
    [tex] 12^1 + 2 \cdot 5^0 = 14 [/tex] , 14/7 = 2

    Next assume:
    [tex] 12^k + 2(5^{k-1}) = 7A [/tex]

    Then, prove for k + 1:
    [tex] 12^{k+1} + 2(5^k) [/tex]

    I can't figure out how to prove this. I know that this can be changed to:
    [tex] 12 \cdot 12^{k} + 2 \cdot 5 (5^{k-1}) [/tex]
    But that doesn't seem to help me much.

    I also tried substituting values for 12^k and 5^(k-1) from above:
    [tex] 12^k = 7A - 2(5^{k-1}) [/tex]
    [tex] 2(5^{k-1}) = 7A - 12^k [/tex]

    This doesn't seem too help either, I can reduce it to:
    [tex] 189A - (12 \cdot 2(5^{k-1})+5(12^k)) [/tex]

    Any suggestions?
    Thanks,
    Tom
     
  2. jcsd
  3. May 10, 2007 #2
    Actually, having come till

    [tex]12.12^k + 2.5(5^{k-1})[/tex],

    the next step should have been

    [tex]7.12^k + 5.12^k + 2.5(5^{k-1})[/tex].
     
  4. May 10, 2007 #3
    Ah, got it now. Thank you. I don't like these induction problems...
     
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