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Math Help - What is the best way to prove this conjecture?

  1. #1
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    What is the best way to prove this conjecture?

    Okay, first post in the forums for me

    This is my first year in high school geometry (i'm a freshman) and I'm loving it! I've run into a spot of trouble though...

    My geometry teacher challenged us with a problem, and if I get it right, he will give me an A for the semester. I don't have the exact conjecture written down, but basically it is a formula to find the nth term in a sequence. For example, this formula could tell you what the 65th term in the sequence 4, 7, 13, 21.... (1st difference is 3, 6, 9, 2nd difference is 3,3,3). I have yet to learn proofs, which is why this is a challenge.

    From the research I have done, I figure that mathematical induction would be the best method. I am guessing that if I can prove it works for a difference of 1, I can prove it for 2,3,4,5, etc. Am I correct in my thinking? Is there a better method? Also, any advice in the process; this is my first time proving a theorem (and he said it IS a theorem, as he has already proven it). Thanks in advance!

    I'll edit this tomorrow when I write down the formula, sorry about its absence right now. If any more information is needed, just ask. I look forward to helping out in the algebra section, and spending some time on these forums!
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  2. #2
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    If the second difference column is constant then the function is a quadratic.
    That is,
    f(n)=an^{2}+bn+c.
    Use the first three numbers to solve for a, b, c.
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  3. #3
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    "...and if I get it right, he will give me an A for the semester." Due to the philosophy of this website, I'll pass on this one.
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  4. #4
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    Quote Originally Posted by wonderboy1953 View Post
    "...and if I get it right, he will give me an A for the semester." Due to the philosophy of this website, I'll pass on this one.
    I expected this to come up, and I suppose I wasn't clear enough. I fully intend to work out the proof myself- once I know how to that is. My teacher said that we could do outside research (as we havn't learned proofs yet) and that is what I am doung. I don't want anyone to solve the problem for me, just explain the process of proving a conjecture using mathematical induction. Hope that clears some stuff up.
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  5. #5
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    More information

    Okay, Here is all of the information, with a better explanation of what I need help on.

    The theorem A_k=[(k-b)*c]+a is used to solve for the nth term in an arithmetic sequence. The variables are as such:
    k=the number term you are looking for (e.g. the 54th term or the 78th term)
    b=the number of terms shown (e.g in the sequence of 2,4,6,8... this value would be 4)
    c=the 1st difference
    a=the last term shown (e.g. 8 in the above example)

    A quick example in case I did not explain it thoroughly:
    Given the sequence 3,6,9,12,15,18,21... determine the 86th term.

    A_k=[(k-b)*c]+a

    A_k=[(86-7)*3]+21

    A_k=[237]+21

    A_k=258

    The 86th term is therefore 258. I understand all of the above. The challenge is to prove this as a theorem, and that is what I am confused about, as I have not yet learned proofs. It is my understanding that if I can prove it to be true when c=0, I can then prove it to be true for n+1. I don't quite understand how to do that, but I can figure it out with time. What I don't get is how to prove this for negative integers. Can anybody offer an explanation at this point? Thanks.
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  6. #6
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    Your general idea about induction is correct. So, you prove the formula is correct for k = 1. Then you prove, under the assumption the formula for A_k is correct, that the formula for A_{k+1} is correct. One addition: you may have to do this for variables other than k. (Or maybe not. You should do something to deserve your A, don't you think?)
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  7. #7
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    I leave my comment purposely obscure because of the whole A incentive.

    If it were me I would replace a with something else.
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  8. #8
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    Ah okay so my thinking is correct. Thats all I needed, really, to get started. Thanks again for all the help here. Quick question; can I mark this thread as 'solved', even if it truely isn't, as I'm not asking for any more help.
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  9. #9
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    Quote Originally Posted by fireballs619 View Post
    Ah okay so my thinking is correct. Thats all I needed, really, to get started. Thanks again for all the help here. Quick question; can I mark this thread as 'solved', even if it truely isn't, as I'm not asking for any more help.
    I see no problem with marking the thread solved; your main problem was "how to get started" and that has been solved.
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