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Thread: Number sequence

  1. #1
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    Number sequence

    5, 10, 17, 26, 37, ...

    I see the pattern, but I can't figure out the equation...any advice?
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  2. #2
    Member u2_wa's Avatar
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    Quote Originally Posted by jzellt View Post
    5, 10, 17, 26, 37, ...

    I see the pattern, but I can't figure out the equation...any advice?
    Hello jzellt:

    You might have noticed that difference between the numbers in sequence corresponds to the odd numbers sequence starting from 5.

    $\displaystyle 5,7,9,11,.....$

    Now we can write it as $\displaystyle 5+0,5+5,5+5+7,5+5+7+9,...$
    We can use this formula for the addition of the odd numbers:
    $\displaystyle \frac{n}{2}(2a+(n-1)*d)$

    Lets plug in the values:
    $\displaystyle \frac{n}{2}(2(5)+(n-1)*2)$
    $\displaystyle \frac{n-1}{2}(10+(n-2)*2)$ (because at n=1, there should be addition of zero i.e 5+0,5+5,....)
    Hence we have the equation: $\displaystyle 5+\frac{n-1}{2}(8+2(n-1))$

    Hope this helps
    Last edited by u2_wa; Mar 13th 2009 at 07:25 AM.
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  3. #3
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    Hello, jzellt!

    I "eyeballed" the solution . . .


    $\displaystyle 5, 10, 17, 26, 37, \hdots$

    Take the differences of consecutive terms.
    Then take the differences of the differences ... and so on.

    . . $\displaystyle \begin{array}{cccccccccc}\text{Sequence} & 5 && 10 && 17 && 26 && 37 \\
    \text{1st diff.} & & 5 && 7 && 9 && 11 \\
    \text{2nd diff.} & & & 2 & & 2 & & 2 \end{array}$


    We see that the second differences are constant.
    This indicates that the generating function is of the second degree, a quadratic.
    . . That is, $\displaystyle f(n)$ contains $\displaystyle n^2.$


    With that in mind, I looked the sequence again, and I saw:

    . . $\displaystyle \begin{array}{c|ccc}
    n & & a_n \\ \hline
    1 & 5&=&2^2+1 \\
    2 & 10 &=&3^2+1 \\
    3 & 17 &=&4^2+1 \\
    4 & 26 &=&5^2+1 \\
    5 & 37 &=&6^2+1 \end{array}$
    . . .$\displaystyle \begin{array}{c|c}\vdots & \vdots \\
    n & \;\;(n+1)^2+1
    \end{array}$


    Therefore: .$\displaystyle f(n) \:=\:(n+1)^2+1 \:=\:n^2+2n+2$

    . . which verifies u2_wa's result.

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