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Math Help - Sequences and sums

  1. #1
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    Sequences and sums

    Hi, I need help with a question regarding sequences and sums:

    The upper of the two following sequences can be entered on the form of a polynomial (ie, an = degree 2 polynomial). Prove that the bottom can not be printed in the form of a degree 2 polynomial. (Hint: start by proving that the top can be entered on the form of a degree 2 polynomial).

    0 1 4 9

    0 1 4 10

    Thanks!
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  2. #2
    MHF Contributor
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    Welcome to the forum.

    It is not exactly clear what "can be entered on the form of a polynomial" means. Do you mean that there exists a polynomial f(x)=ax^2+bx+c and four points x_1,x_2,x_3,x_4 such that f(x_1)=0, f(x_2)=1, f(x_3)=4, f(x_4)=9? Is there any condition on x_i, for example: x_1=0, ..., x_4=3 or x_3=x_2+1?
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  3. #3
    MHF Contributor

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    The quadratic function, f, such that f(0)= 0, f(1)= 1, f(2)= 4, f(3)= 9 is fairly obvious. If we are to assume that the second function must give f(0)= 0, f(1)= 1, f(2)= 4, f(3)= 10, then the answer is simple since a quadratic function, having 3 coefficients, is determined by 3 equations.
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