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Math Help - Proof by Induction (2n+1)(7^n)-1=4x

  1. #1
    LHS
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    Proof by Induction (2n+1)(7^n)-1=4x



    I would be very grateful if anyone can help
    I think I have a way, but i'm a little dubious about it. It involves subtracting f(n) from f(n+1), and factorising out a 4
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  2. #2
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    hint on the induction step:

    (2(n+1)+1)7^{n+1}-1 = (2n+1)7^{n+1} +2\cdot 7^{n+1} -1 = (2n+1)7^n + (2n+1)7^n\cdot 6 +2\cdot 7^{n+1} -1 =

    = (2n+1)7^n -1 +(2n+1)7^n\cdot 6+ 2\cdot 7^n\cdot(3+4) = (2n+1)7^n -1 +(2n+1)7^n\cdot 6+ 7^n\cdot 6 +8\cdot 7^n =

    = (2n+1)7^n -1 +(2n+2)7^n\cdot 6 +8\cdot 7^n

    can you follow from here?
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  3. #3
    LHS
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    Ok, so what you have shown there is that:
    f(n+1) = f(n) + 4[7^n(3n+5)]

    So you are saying that if f(n+1) equals the original function, add a multiple of 4?
    How does that prove that f(n) is a multiple of 4?
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  4. #4
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    do you know how proofs by induction work? Mathematical induction - Wikipedia, the free encyclopedia

    1. base case: n=1 ... f(1) = 20 is divisible by 4, base case verified.
    2. inductive step: suppose the statement holds for some n. show that the statement holds for n+1. And that is what i've done in the previous post.
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  5. #5
    LHS
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    Sorry I'm being retarded. Yes I am meant to know how induction works :P I can do the simpler questions, this through me off as it had a n within the bracket as well as the power.

    Yes, that makes perfect sense now! Thanks for your help, this has been bugging me!
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