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Thread: Sequence

  1. #1
    Senior Member
    Jan 2009


    How do I find the general term a(n)?

    2, 8, 18, 32, 50

    I know a(n)=$\displaystyle 2n^2$

    But how do I get that?

    I know you find the first order difference, b(n)

    b(n)= 6, 10, 14, 18 = $\displaystyle 4n-2$

    $\displaystyle \sum_{k=1}^{n-1} 4k-2 $

    $\displaystyle =2+2n(n-1)-2(n-1)$

    Obviously I messed up on that last step. How do I correctly do it? Thanks.
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  2. #2
    Like a stone-audioslave ADARSH's Avatar
    Aug 2008
    I don't really understood how you are doing but this is the way I do it

    $\displaystyle S= 2+8+18+32+.....T_n$

    $\displaystyle S= 0+2+8+18+....T_{n-1} + T_n$

    Subtract the above two equations

    $\displaystyle S-S = 2 +6+10+14+18+...+(T{n}-T_{n-1})~ - T_n$


    T_n = 2 +6 +10+ 14+18+...(T{n}-T_{n-1})$


    T_n= \text{Sum of the n terms of AP with first term 2 and common difference 4}$

    $\displaystyle T_n = \frac{n(2\times 2+(n-1)4)}{2}$

    $\displaystyle T_n=\frac{n(4n)}{2}$

    $\displaystyle T_n= 2n^2$

    we took first order difference of the complete sequence
    and then the last term could be easily found by using sum of n terms of Arithmetic Progression
    Last edited by mr fantastic; Mar 5th 2009 at 03:44 AM. Reason: Requested edit of small mistake
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  3. #3
    MHF Contributor

    Apr 2005
    Oh, that's very clever!

    chengbin, there is no "formula" for this- you just have try whatever you can think of and look for patterns. Since this sequence was neither an arithmetic sequence nor a geometric sequence, (the "easy" sequences), Adarsh tried looking at the "first difference" (the difference between consecutive terms) and saw that it was an arithmetic sequence (the "second difference" was constant).

    I, seeing that all the numbers in the sequence are even, would have been inclined to divide each term, giving 1, 4, 9, 25, ... and recognized those as squares. Since $\displaystyle a_n/2= n^2$, $\displaystyle a_n= 2n^2$.
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  4. #4
    Senior Member
    Jan 2009
    Thanks for the answer.

    I remember doing this when I did sequences. I don't remember exactly. But thanks.
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