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Math Help - help me with Inductive Proof #2

  1. #1
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    help me with Inductive Proof #2

    n^2-1 is divisible by 8 whenever n is an odd positive integer

    1) Basic step: P(1) = 1-1 =0 is divisible by 8.

    2) inductive step:
    hypothesis: P(k): k^2-1 is divisible by 8

    conclusion: P(k+2): (k+2)^2 -1 is divisible by 8?
    OR
    P(k+1): (k+1)^2 -1 is divisible by 8? <- to make it odd

    help me pls
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  2. #2
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    Quote Originally Posted by zpwnchen View Post
    n^2-1 is divisible by 8 whenever n is an odd positive integer

    1) Basic step: P(1) = 1-1 =0 is divisible by 8.

    2) inductive step:
    hypothesis: P(k): k^2-1 is divisible by 8

    conclusion: P(k+2): (k+2)^2 -1 is divisible by 8?
    OR
    P(k+1): (k+1)^2 -1 is divisible by 8? <- to make it odd

    help me pls

    It is P(k+2), of course: if k is an odd positive integer, the next ODD integer is k + 2, not k + 1.

    Without induction: n^2 - 1 = (n-)(n+1). As n is odd, exactly one of n-1 or n+1 is divisible by 4 and the other factor is even ==> the whole thing is divisible by 4*2 = 8.

    Tonio
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