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Homework Help: Nth term of a sequence

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Is this statement correct?

    If the nth term of a sequence is a quadratic expression in n, then the sequence is an A.P.


    2. Relevant equations



    3. The attempt at a solution

    Take arbitrary t(n)=n^2-2n-2
    I substituted 1,2,3 in the above expression and noted the c.d.
    It is not constant.
    But the book says that this statement is correct.
    Any ideas?
     
  2. jcsd
  3. Sep 26, 2010 #2

    Mark44

    Staff: Mentor

    The book's statement doesn't seem correct to me. The simplest sequence that fits the description is an = {n2} = {1, 4, 9, 16, ..., n2, ...} This is definitely not an arithmetic sequence for the reason you stated - the difference between pairs of successive terms is not constant.
     
  4. Sep 26, 2010 #3
    Thanks!
     
  5. Sep 26, 2010 #4

    Office_Shredder

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    If the terms of a sequence are from a quadratic formula, then the difference between the nth and (n-1)st terms form an arithmetic progression.

    For example if the sequence is 1,4,9,16,25,...

    then the differences are

    4-1, 9-4, 16-9, 25-16,...
    3,5,7,9,...

    That might be what they meant to refer to
     
  6. Sep 26, 2010 #5
    It is true only in case on n^2 (may be in some other cases too).
    But if you take an expression like n^2 +2n-1, then Tn- T(n-1)=2n is not independent of n i.e. it is not a constant.
     
  7. Sep 26, 2010 #6

    Office_Shredder

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    The sequence whose nth term is 2n is an arithmetic progression
     
  8. Sep 26, 2010 #7
    You did not get me. In my expression, 2n is the difference between two consecutive terms of the sequence. It is not the nth term.
    If you take nth term of the sequence as 2n, it violates the question as it is not a quadratic expression.
     
  9. Sep 26, 2010 #8

    Office_Shredder

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    What I said is that the sequence whose nth term is the difference between consecutive terms of the quadratic sequence is an arithmetic progression. So for your example, the nth term of the sequence I'm describing is T(n)-T(n-1), and this new sequence is an arithmetic progression
     
  10. Sep 27, 2010 #9

    HallsofIvy

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    If the sum (form i= 1 to n) of a sequence of numbers is a quadratic function of n, then the sequence is arthmetic.
     
  11. Sep 27, 2010 #10
    Can you give an example?
     
  12. Sep 27, 2010 #11

    Mark44

    Staff: Mentor

    The sum of the first n integers.
    [tex]\sum_{k= 1}^n k = \frac{n(n + 1)}{2}[/tex]
     
  13. Sep 27, 2010 #12
    Thanks!
     
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