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Math Help - help me with Inductive Proof #1

  1. #1
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    help me with Inductive Proof #1

    Here is the given:
    n^2 -7n+12 is non-negative integer whenever n is an integer with n \geq 3

    Here is what i come up with:
    Hypothesis: P(k) = k^2 - 7k + 12
    Conclusion: P(k+3) = (k+3)^2 - 7(k+3) + 12 > 0
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  2. #2
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    Try the base case with n=3.

    3^{2}-7(3)+12=0

    Now, try the induction step:

    (k+1)^{2}-7(k+1)+12

    =k^{2}+2k+1-7k-7+12

    Rearrange and group:

    =(\underbrace{k^{2}-7k+12}_{\text{this is 0}})+(2k+1-7)

    \geq 0+2k+1-7=2k-6

    \geq 2(3)-6=0

    Since n=3 gives 0, then any number larger than 3 will give a result larger than 0 and is non negative.
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