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  1. #1 Solving an Equation 
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    How would one go about solving this equation for a:



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  3. #2  
    Forum Radioactive Isotope MagiMaster's Avatar
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    I'd start by finding a closed form for the sum (one exists), and go from there.


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  4. #3  
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    Quote Originally Posted by MagiMaster
    I'd start by finding a closed form for the sum (one exists), and go from there.
    In terms of , but that would be rather tedious.
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  5. #4  
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    Check out the Wikipedia article http://en.wikipedia.org/wiki/Arithmetic_series.
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  6. #5 Re: Solving an Equation 
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    Quote Originally Posted by Ellatha
    How would one go about solving this equation for a:

    Start with



    This formula can be proved by induction or by the following trick
    Code:
    1              2            3     ...     N
    N             N-1         N-2  .....      1
    --------------------------------------------
    (N+1) +     (N+1) +      (N+1)  + ...+ (N+1)
    You should be able to use this to solve your problem.
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  7. #6 Re: Solving an Equation 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Ellatha
    How would one go about solving this equation for a:

    Start with



    This formula can be proved by induction or by the following trick
    Code:
    1              2            3     ...     N
    N             N-1         N-2  .....      1
    --------------------------------------------
    (N+1) +     (N+1) +      (N+1)  + ...+ (N+1)
    You should be able to use this to solve your problem.
    But that is for the case that , but for my formula, it is not , but . Also, n does not equal 1, but 99. Would I need to manipulate the formula?
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  8. #7  
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    Yes. Try a verbal description of the formula. The series 2+3+4+5 is the same as (5+2)+(3+4) [associative and commutative], however, the second representation has the advantage that each term is the same (i.e. 7). This can be done with the sum of any series of consecutive numbers. Changing n to n+1 just shifts where the series begins and ends. If you are adding X numbers, then there are X/2 pairs of numbers with the same value. In your formula 100 is the first number in the series (99+1) and a+1 is the last number in the series. So all you have to do is figure out how many pairs of number there are in the series, and what the value of each of the pairs are (which is simply the first number in the series plus the last number in the series) and you are good to go.
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  9. #8 Re: Solving an Equation 
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by DrRocket
    Quote Originally Posted by Ellatha
    How would one go about solving this equation for a:

    Start with



    This formula can be proved by induction or by the following trick
    Code:
    1              2            3     ...     N
    N             N-1         N-2  .....      1
    --------------------------------------------
    (N+1) +     (N+1) +      (N+1)  + ...+ (N+1)
    You should be able to use this to solve your problem.
    But that is for the case that , but for my formula, it is not , but . Also, n does not equal 1, but 99. Would I need to manipulate the formula?
    You you still need to a little bit of work. But it should be pretty easy.

    For instance



    and

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  10. #9 Re: Solving an Equation 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Ellatha
    Quote Originally Posted by DrRocket
    Quote Originally Posted by Ellatha
    How would one go about solving this equation for a:

    Start with



    This formula can be proved by induction or by the following trick
    Code:
    1              2            3     ...     N
    N             N-1         N-2  .....      1
    --------------------------------------------
    (N+1) +     (N+1) +      (N+1)  + ...+ (N+1)
    You should be able to use this to solve your problem.
    But that is for the case that , but for my formula, it is not , but . Also, n does not equal 1, but 99. Would I need to manipulate the formula?
    You you still need to a little bit of work. But it should be pretty easy.

    For instance



    and

    Alright, here is my work:











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  11. #10 Re: Solving an Equation 
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    Quote Originally Posted by Ellatha

    Alright, here is my work:











    Close



    Should be



    Which still gets you

    But is supposed to be an integer, so there really is no exact solution.

    You can calculate that



    and

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