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Math Help - Mathematical Induction

  1. #1
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    Mathematical Induction

    Question:

    Use the principal of Mathematical Induction to prove 2|(n2+n) for all n>=0
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by oldguy View Post
    Question:

    Use the principal of Mathematical Induction to prove 2|(n2+n) for all n>=0
    for n=0, 2|(0^2 + 0) = 2|0

    suppose it is true that for n=k, 2|(k^2 + k).
    show that if n=k+1, then 2|((k+1)^2 + (k+1)).

    now, (k+1)^2 + (k+1) = k^2 + 2k + 1 + k + 1 = k^2 + k  + 2k + 2 = k^2 + k  + 2(k + 1)..
    notice that, 2|k^2 + k and 2|2(k + 1)

    therefore, 2|k^2 + k  + 2(k + 1) or 2|(k+1)^2 + (k+1) . QED
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  3. #3
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    Hello, oldguy!

    Use Mathematical Induction to prove: . 2\,|\,(n^2+n) for all n \geq 0
    S(n)\!: \;n^2+n is a multiple of 2.


    Verify S(1)\!:\;\;1^2 + 1 \:=\:2 . . . True!


    Assume S(k)\!:\;\;k^2 + k \:=\:2a for some integer a


    Add 2k+2 to both sides: . k^2 + k + {\color{blue}2k + 2} \;=\;2a + {\color{blue}2k + 2}

    We have: . k^2 + 2k + 1 + k + 1 \;=\;2a + 2k + 2

    . . . . . . . .. \underbrace{(k+1)^2 + (k+1)}_{\text{Left side of S(k+1)}}   \;=\;\underbrace{2(a + k + 1)}_{\text{multiple of 2}}


    Therefore, we have proved S(k+1).
    . . The inductive proof is complete.

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  4. #4
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    Thanks for the help.
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