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Math Help - Math induction explanation

  1. #1
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    Math induction explanation

    I have a question and have set up a formula, but I am not sure how to do the induction steps. Each time I try I do it wrong. Can someone help? Here is the question:
    Create a proof by mathematical induction that demonstrates that the sum of the first n even numbers is equal to n(n + 1).
    Here is my equation:

    2+4+6++n=n(n+1)

    And what I have come up with:

    The first step is 2 = 1(1+1); 2=2 which is true.

    Now we assume that f(n)=f(n+1) is true


    2+4+6++n+(n+1)=n(n+1)+(n+1)

    Ok, so when I solve it it doesn't work. What did I do wrong???
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by rbotcaldwell View Post
    I have a question and have set up a formula, but I am not sure how to do the induction steps. Each time I try I do it wrong. Can someone help? Here is the question:
    Create a proof by mathematical induction that demonstrates that the sum of the first n even numbers is equal to n(n + 1).
    Here is my equation:

    2+4+6++n=n(n+1) Be careful!! The n'th even number is 2n, not n.

    And what I have come up with:

    The first step is 2 = 1(1+1); 2=2 which is true.

    Now we assume that f(n)=f(n+1) is true


    2+4+6++n+(n+1)=n(n+1)+(n+1) Should be 2+4+6++2n+2(n+1)=n(n+1)+2(n+1).

    Ok, so when I solve it it doesn't work. What did I do wrong???
    Try taking it from there.
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  3. #3
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    Huhh?

    It still doesnt work. Simplify your equation and it becomes (n+1)(n+2). When you plug into the original equation s(2)=2+4=(2+1)(2+2); 6 does not equal 12. I am so not understanding this and I am sure I am making it harder than needs be!
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by rbotcaldwell View Post
    ...Simplify your equation and it becomes (n+1)(n+2)....
    exactly! which proves the claim

    that's what you get when you replace n with (n + 1) in n(n + 1)

    you have shown f(n) => f(n + 1)


    also, assuming f(n) = f(n + 1) is not a part of mathematical induction
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  5. #5
    Junior Member NoFace's Avatar
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    Quote Originally Posted by rbotcaldwell View Post
    Now we assume that f(n)=f(n+1) is true

    2+4+6++n+(n+1)=n(n+1)+(n+1)

    Ok, so when I solve it it doesn't work. What did I do wrong???
    What has helped me is to visualize the goal. Just write it out, and work towards it. You always want to be able to manipulate your formula to get f(n+1) in the end, if that's the route you're going.

    So you say: f(n) implies f(n+1), so in this case your f(n+1) will look like this:

    (n+1)(n+2)

    which is what you got. So what you are doing by plugging in 2 to f(n+1) is actually getting the answer for 2+1=3, which is 6=1+2+3, which is true.

    Induction kind of seems like a hat trick at first, but try proving something with induction that doesn't work for anything but the base case. You'll find it quite impossible.
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