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Math Help - How to prove that n^2 + n + 2 is even??

  1. #1
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    How to prove that n^2 + n + 2 is even??

    using mathematical induction...
    help please

    thank you
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  2. #2
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    It's a pity to use induction
    n+n+2 = (n+1) - n + 1
    n odd => (n+1) even => (n+1) - n + 1 even
    n even => (n+1) odd => (n+1) - n + 1 even

    By induction
    Easy for n=0
    If right for n then n+n+2 even
    (n+1)+(n+1)+2 = n+2n+1 + n + 3 = n+n+2 +2n+2= n+n+2 +2(n+1)
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  3. #3
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    Hello, NeedHelp18!

    Prove by induction: . n^2 + n + 2 is even.

    Verify S(1)\!:;\;1^2 + 1 + 2 :=\:4 . . . even!


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


    Add 2k+2 to both sides: . (k^2+k+2) + (2k+2) \;=\;2a + (2k+2)

    We have: . k^2+3k + 4 \;=\;2a + 2k+2

    . . (k^2 + 2k + 1) + (k + 1) + 2 \;=\;2a + 2k + 2

    . . \underbrace{(k+1)^2 + (k+1) + 2}_{S(k+1)} \;=\;\underbrace{2(a + k + 1)}_{\text{even}}


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

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  4. #4
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    n^{2}+n+2=\frac{4n^{2}+4n+8}{4}=\frac{(2n+1)^{2}+7  }{4}.

    2n+1 is odd hence (2n+1)^2 is odd and (2n+1)^{2}+7 is even, therefore n^{2}+n+2 is even.
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