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Math Help - mathematical induction

  1. #1
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    mathematical induction

    use mathematical induction to prove the formula for every positive integer n.

    3 + 5 + 7 + ... + (2n +1)= n(n +2)


    this is a beastly problem.

    i have proven that

    n=1 so 2(1)+1=1(1)+2
    3=3 so this works but now im dont know what to do next. i know im suppose to go further but im so lost. please help me understand this.
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  2. #2
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    Skip the trivial case and assume true for

    1+3+5+......+(2k+1)=k(k+1)

    Show P_{k+1} is true:

    1+3+5+....+(2k+1)+(2k+3)=k(k+2)+2k+3=(k+1)(k+3)

    Therefore, it is true for P_{k+1}
    Last edited by galactus; September 26th 2009 at 12:28 PM.
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  3. #3
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by flexus View Post
    use mathematical induction to prove the formula for every positive integer n.

    3 + 5 + 7 + ... + (2n +1)= n(n +2)


    this is a beastly problem.

    i have proven that

    n=1 so 2(1)+1=1(1)+2
    3=3 so this works but now im dont know what to do next. i know im suppose to go further but im so lost. please help me understand this.
    Assume true for n, now you have to prove it for n+1, i.e. that:
    3+5+7+...+(2n+1) + (2(n+1)+1) = (n+1)(n+3)

    And you'd do that by saying:

    3+5+7+...+(2n+1) + (2(n+1)+1) = n(n+2) + (2(n+1)+1)

    and working from there.

    Because you have *assumed* that it's true for n, you can directly substitute the n(n+2) term for the series up to (2n+1).

    And having *assumed* that it's true for n, (that bit's called the "induction hypothesis"), you will have *proved* that it still holds for n+1 (that bit's called the "induction step" and it's usually the one where the algebra happens).

    So once you've shown it true for the base case, and by assuming the truth of the induction hypothesis for n you've shown it still holds for n+1, "the result follows by the Principle of Mathematical Induction".
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  4. #4
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by galactus View Post
    Skip the trivial case and assume true for

    1+3+5+......+(2k+1)=k(k+1)

    Show P_{k+1} is true:

    1+3+5+....+(2k+1)+(2k+3)=k(k+1)+2k+3=(k+1)(k+3)

    Therefore, it is true for P_{k+1}
    In your second line of maths I think you mean k(k+2) not k(k+1).
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  5. #5
    Eater of Worlds
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    Quote Originally Posted by Matt Westwood View Post
    In your second line of maths I think you mean k(k+2) not k(k+1).
    Yes, that was a typo. Thanks for the heads up.
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  6. #6
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    Quote Originally Posted by galactus View Post
    Skip the trivial case and assume true for

    1+3+5+......+(2k+1)=k(k+1)

    Show P_{k+1} is true:

    1+3+5+....+(2k+1)+(2k+3)=k(k+2)+2k+3=(k+1)(k+3)

    Therefore, it is true for P_{k+1}
    lol man your avatar is funny but also crazy haha
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