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Math Help - Need help finding the pattern in this sequence of numbers

  1. #1
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    Need help finding the pattern in this sequence of numbers

    I wasn't sure where to place this problem in the forums. I hope this area is OK. I'm trying to find a pattern in this series of numbers. How can I find a simplified expression, where I evaluate it at my given n, and the proper f(n) is returned?

    n=4, f(4)=6
    n=5, f(5)=10
    n=6, f(6)=15
    n=7, f(7)=21

    I found that the function can be represented by the sum \sum_{k=1}^{n-1}{(n-k)}.

    So for n=4, f(4)=3+2+1
    n=5, f(5)=4+3+2+1
    ...

    Is there a way I can find a simplified expression? For example, the series
    2,4,8,16 can be represented by n^2. I was curious if I can find a simplified expression for this pattern.

    Thanks

    Edit: This should probably be moved to the number theory forum. Sorry
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  2. #2
    Lord of certain Rings
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     f(n) = 1 + 2 + 3 + . . . . . .+ (n-2)  + (n-1) = \frac{n(n-1)}2
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Fourier View Post
    I wasn't sure where to place this problem in the forums. I hope this area is OK. I'm trying to find a pattern in this series of numbers. How can I find a simplified expression, where I evaluate it at my given n, and the proper f(n) is returned?

    n=4, f(4)=6
    n=5, f(5)=10
    n=6, f(6)=15
    n=7, f(7)=21

    I found that the function can be represented by the sum \sum_{k=1}^{n-1}{(n-k)}.

    So for n=4, f(4)=3+2+1
    n=5, f(5)=4+3+2+1
    ...

    Is there a way I can find a simplified expression? For example, the series
    2,4,8,16 can be represented by n^2. I was curious if I can find a simplified expression for this pattern.

    Thanks

    Edit: This should probably be moved to the number theory forum. Sorry
    Look at the difference table:

    Code:
    6   10   15   21
      4    5    6
        1    1
    That the second differences are constant tells you that the series can be
    generated by a quadratic in n.

    Now just fit a general quadratic f(n)=an^2+bn+c to the data.

    RonL
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Fourier View Post
    I wasn't sure where to place this problem in the forums. I hope this area is OK. I'm trying to find a pattern in this series of numbers. How can I find a simplified expression, where I evaluate it at my given n, and the proper f(n) is returned?

    n=4, f(4)=6
    n=5, f(5)=10
    n=6, f(6)=15
    n=7, f(7)=21

    I found that the function can be represented by the sum \sum_{k=1}^{n-1}{(n-k)}.

    So for n=4, f(4)=3+2+1
    n=5, f(5)=4+3+2+1
    ...

    Is there a way I can find a simplified expression? For example, the series
    2,4,8,16 can be represented by n^2. I was curious if I can find a simplified expression for this pattern.

    Thanks

    Edit: This should probably be moved to the number theory forum. Sorry
    Plus this is the well known triangle numbers whos generating function is \frac{n(n+1)}{2}
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