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Math Help - Mathematical Inuction problem

  1. #1
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    Mathematical Inuction problem

    Having problems with this question

    n^3 - 7n + 3 is divisible by 3, for each integer n => 0.

    ---------------My work-------------------------

    Show n is true for n = 0

    0^3 -7(0) + 3 = 3

    Then show n is true for n = K +1

    (K + 1)^3 -7(k + 1) + 3 = k^3 + 3k^2 + 3k +1 - 7K -7 = k^3 + 3k^2 -4k -3

    Im kinda stuck on how to proceed...
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  2. #2
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    Quote Originally Posted by Thetheorycase View Post
    Having problems with this question

    n^3 - 7n + 3 is divisible by 3, for each integer n => 0.

    ---------------My work-------------------------

    Show n is true for n = 0

    0^3 -7(0) + 3 = 3

    Then show n is true for n = K +1

    (K + 1)^3 -7(k + 1) + 3 = k^3 + 3k^2 + 3k +1 - 7K -7 = k^3 + 3k^2 -4k -3

    Im kinda stuck on how to proceed...
    k^3 + 3k^2 -4k=(k-1) k (k+4)
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  3. #3
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    Quote Originally Posted by Thetheorycase View Post
    Having problems with this question

    n^3 - 7n + 3 is divisible by 3, for each integer n => 0.

    ---------------My work-------------------------

    Show n is true for n = 0

    0^3 -7(0) + 3 = 3

    Then show n is true for n = K +1

    (K + 1)^3 -7(k + 1) + 3 = k^3 + 3k^2 + 3k +1 - 7K -7 +3 = k^3 + 3k^2 -4k -3 no, you are oversimplifying!

    Im kinda stuck on how to proceed...
    You had a small typo (omitted +3).

    Continuing...

    k^3+3k^2+3k+1-7k-7+3=\left(k^3-7k+3\right)+3k^2+3k+1-7

    =\left(k^3-7k+3\right)+3k^2+3k-6

    Now, what can we conclude ?
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  4. #4
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    Quote Originally Posted by Archie Meade View Post
    You had a small typo (omitted +3).

    Continuing...

    k^3+3k^2+3k+1-7k-7+3=\left(k^3-7k+3\right)+3k^2+3k+1-7

    =\left(k^3-7k+3\right)+3k^2+3k-6

    Now, what can we conclude ?
    We conclude that n^3 - 7n + 3 is divisible by 3 meaning: n^3 - 7n + 3 = 3r for some int r

    I dont know where i Should sub it?
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  5. #5
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    Quote Originally Posted by Thetheorycase View Post
    We conclude that n^3 - 7n + 3 is divisible by 3 meaning: n^3 - 7n + 3 = 3r for some int r

    I dont know where i Should sub it?
    Induction tries to find out if the hypothesis being true for any n=k
    causes the hypothesis to be true for n=k+1.

    True for n=1 causes true for n=2
    True for n=2 causes true for n=3
    True for n=3 causes true for n=4 ...... to infinity.

    Using k and k+1 examines this cause and effect in general.
    If true for k causes true for k+1,
    then we only need to examine the first n, as the hypothesis would then be true for all n.

    So, if the formula is valid for some n=k,
    and being valid for n=k causes validity for n=k+1,
    you then only need to test the first term.

    This is why you should write the k+1 expression containing the k expression.

    If 3 does divide k^3-7k+3, then it will definately divide \left(k^3-7k+3\right)+3\left(k^2+k-2\right)
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