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Math Help - sequence formula

  1. #1
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    sequence formula

    I need find a closed form for this recurrence relation:

    b sub n = b sub n-1 + n^2 for n >0 and b sub 0 = 3

    Anybody know how to do this?
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  2. #2
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    Quote Originally Posted by jzellt View Post
    I need find a closed form for this recurrence relation:

    b sub n = b sub n-1 + n^2 for n >0 and b sub 0 = 3

    Anybody know how to do this?
    There is probably a more elegant way than this:

    1. Calculate the first five values
    2. Calculate the differences between consecutive values
    3. Repeat this process until you get a constant sequence.
    4. Determine the number of repetitions. That is the degree of the closed form:
    Code:
    degree 3:  3    4    8     17    33    ...
    degree 2:     1    4    9     16    ...
    degree 1:       3    5     7    ...
    degree 0:         2     2
    That means you are looking for a term with degree 3:

    s_n = a\cdot n^3+b\cdot n^2+c\cdot n + d

    From the initial row you know:
    s_0 = 3
    s_1 = 4
    s_2 = 8
    s_3 = 17

    Thus you have to solve a system of simultaneous equations:
    \left|\begin{array}{r}d=3\\a+b+c+d = 4\\8a+4b+2c+d=8\\27a+9b+3c+d=17\end{array}\right.

    After a few steps you'll get:

    s_n = \frac13 n^3 + \frac12 n^2 + \frac16n+3
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