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Math Help - Induction help

  1. #1
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    Induction help

    Hi I need help in regards with proving this problem.

    If the sequence is 1,13,37,73,121
    then it should follow like this 1+12+24+36......12n+(12n+12)
    For the formula 6n^2 - 6n + 1.

    How would I go on to prove this with induction exactly? Thanks
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  2. #2
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    Quote Originally Posted by monkbear View Post
    Hi I need help in regards with proving this problem.
    If the sequence is 1,13,37,73,121
    then it should follow like this 1+12+24+36......12n+(12n+12)
    For the formula 6n^2 - 6n + 1.
    How would I go on to prove this with induction exactly? Thanks
    Have a look at this Page
    Last edited by Plato; May 26th 2011 at 02:17 PM.
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  3. #3
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    Quote Originally Posted by monkbear View Post
    How would I go on to prove this with induction exactly? Thanks
    Induction works in two steps. First you prove it true for some base case. In this case, if P(n) reads

    1 + 12 + 24 + ... + 12n + (12n + 12) = 6n^2 - 6n + 1

    show that P(0) is true. That should be straight-forward computation. The inductive step is to assume it is true for some k \geq 0. Assuming P(k), demonstrate P(k+1). By establishing those two facts regarding P(n), you can prove it is true for every n\geq 0. Do you understand why this two-step process works? In other words, do you know why the principle of mathematical induction (PMI) is a valid rule of inference?

    Quote Originally Posted by Plato View Post
    Have a look at this page.
    That link didn't work for me Plato (using FF2). Broken?
    Last edited by bryangoodrich; May 26th 2011 at 02:11 PM. Reason: opps, forgot equality
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  4. #4
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    Quote Originally Posted by bryangoodrich View Post
    that link didn't work for me plato (using ff2). Broken?
    fixed.
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  5. #5
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    I've never used Wolfram Alpha before, but I have to say ... that's awesome!
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  6. #6
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    Quote Originally Posted by monkbear View Post
    Hi I need help in regards with proving this problem.

    If the sequence is 1,13,37,73,121
    then it should follow like this 1+12+24+36......12n+(12n+12)
    For the formula 6n^2 - 6n + 1.

    How would I go on to prove this with induction exactly? Thanks
    You should have a closer look at the pattern.

    u_1=1=1+12(0)

    u_2=13=1+12(1)

    u_3=37=1+12(1)+12(2)

    u_n=1+12(1)+12(2)+....+12(n-1)

    To prove using induction that

    u_n=1+12(1)+12(2)+....+12(n-1)=6n^2-6n+1


    P(k)

    1+12(1)+12(2)+....+12(k-1)=6k^2-6k+1

    P(k+1)

    1+12(1)+12(2)+.....+12(k-1)+12k=6(k+1)^2-6(k+1)+1\;\;\;?

    Try to show that IF P(k) is true THEN P(k+1) will also be true.

    Proof

    If P(k) is true, then P(k+1) is

    1+12(1)+12(2)+...+12(k-1)+12k=6k^2-6k+1+12k=6k^2+6k+1

    which compares to

    6(k+1)^2-6(k+1)+1=6\left(k^2+2k+1\right)-6k-6+1

    =6k^2+12k+6-6k-6+1

    =6k^2+6k+1

    It remains to test the base case for n = 1, which is clearly true.
    Last edited by Archie Meade; May 27th 2011 at 10:47 AM. Reason: typo
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