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Math Help - Recursion help.

  1. #1
    Junior Member
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    Recursion help.

    I have a general question about recursion. When you're either evaluating a recursive definition like:

    f(0) = 1, f(1) = 0, f(2) = 2, f(n) = 2f(n-3) for n\geq3

    and trying to find a general formula for f(n)

    or trying to give a recursive definition for a formula like:

    a_n = n(n+1)

    Is there an easier way to do it then look at the sequence and try to find some pattern between the 2 formulas, because I'm really bad at figuring about patters in sequences and forming formulas from them.

    Any help would be greatly appreciated. Thank you.
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  2. #2
    MHF Contributor

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    In answering this, here is something I do not usually do: give a detailed answer.
    But it may help you to see how we work on problems.
    I used a computer algebra system to produce some 15 terms of the example.
    \begin{array}{*{20}c}   n &\vline &  0 &\vline &  1 &\vline &  2 &\vline &  3 &\vline &  4 &\vline &  5 &\vline &  6 &\vline &  7 &\vline &  8  \\<br />
\hline   {a_n } &\vline &  1 &\vline &  0 &\vline &  2 &\vline &  2 &\vline &  0 &\vline &  4 &\vline &  4 &\vline &  0 &\vline &  8  \\ \end{array}

    From that brief table, we see blocks of three and powers of two.
    It then took some time to find a closed form of the sequence.
    Here is what I found using the ceiling function:
     a_n  = \left\{ {\begin{array}{cl}<br />
   {0} & {,3\text{ divides }\left( {n - 4} \right)}  \\<br />
   {2^{\left\lceil {\displaystyle\frac{{n - 1}}<br />
{3}} \right\rceil } } & .\text{else}  \\ \end{array} } \right.
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