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Higher IB induction proof - what am I missing??

Discussion in 'Mathematics' started by beedge, Dec 21, 2011.

  1. beedge

    beedge New commenter

    I've been brushing up my all-round skills since we broke up, and I'm come across a question I just cannot do; namely to prove (using induction) that
    7^n - 4^n - 3^n is divisible by 12
    It starts off easily enough:
    7^k - 4^k - 3^k = 12A
    Then considering:
    7^k+1 - 4^k+1 - 3^k+1, which of course is:
    7(7^k) - 4(4^k) - 3(3^k) If you substitute for 7^k, you end up with:
    84A + 3(4^k) + 4(3^k) which is where I'm stuck.
    Ahhh...... The process of typing my thoughts has helped me to spot something I haven't spotted the last half an hour. Can you start converting the 3(4^k) into a 12(4^k-1)? Is that allowed?
     
  2. beedge

    beedge New commenter

    I've been brushing up my all-round skills since we broke up, and I'm come across a question I just cannot do; namely to prove (using induction) that
    7^n - 4^n - 3^n is divisible by 12
    It starts off easily enough:
    7^k - 4^k - 3^k = 12A
    Then considering:
    7^k+1 - 4^k+1 - 3^k+1, which of course is:
    7(7^k) - 4(4^k) - 3(3^k) If you substitute for 7^k, you end up with:
    84A + 3(4^k) + 4(3^k) which is where I'm stuck.
    Ahhh...... The process of typing my thoughts has helped me to spot something I haven't spotted the last half an hour. Can you start converting the 3(4^k) into a 12(4^k-1)? Is that allowed?
     
  3. 7*7^n-4*4^n-3*3^n = 7 (7^n-4^n-3^n) + ......+ ........
     
  4. Quite a sneaky one for IB.
    Is this an actual IB question, or just found in a textbook.
     
  5. I was about to go out as I made my last post. Now that I look properly at your post I see that you had in fact solved the problem.

    If you want another, you might like this one.

    Prove that 5^(2n+2)-24n-25 is divisible by 576 for n=1,2,3.....
     
  6. beedge

    beedge New commenter

    yes in a text book - definitely harder than typical divisibility induction questions
     
  7. beedge

    beedge New commenter

    Null,
    I'm not sure what you were doing in your first post. However, from you second post, does that mean that the following is correct:
    84A + 3(4^k) + 4(3^k)
    = 84A + 12(4^k-1) + 12(3^k-1)
    = 12 (7A + 4^k-1 + 3^k-1)
    And that's it?
    I will do my best that other problem you set. Could take a while...
     
  8. Colleen_Young

    Colleen_Young Occasional commenter

    beedge that's what I got - and so did some other chap on a forum I found later!
     
  9. Sorry about that.


    7*7^n-4*4^n-3*3^n = 7 (7^n-4^n-3^n) + ......+ ........


    was short for


    7*7^n-4*4^n-3*3^n = 7 (7^n-4^n-3^n) + 3*4^n + 4*3^n.


    patronisingly (?) leaving the gaps for you to fill in.

    As you see it was what you'd already done!
     
  10. beedge

    beedge New commenter

    Prove that 5^(2n+2)-24n-25 is divisible by 576 for n=1,2,3.....
    Did it! What an elegant little proof. My first thought was "this looks horendous" but it worked out very nicely. Do you have any other good ones?
     

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