Results 1 to 2 of 2

Thread: General Term for a Sequence

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    144

    General Term for a Sequence

    Find an of {an} given a1 = 1, and an+1 = Sn + 2n + 1
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    849
    Hello, MATNTRNG!


    I hope the $\displaystyle S$ is a typo . . .


    $\displaystyle \text{Find }a_n\text{ of }\{a_n\}.\;\text{ given: }\:a_1 = 1,\;\;a_{n+1} = a_n + 2n + 1$

    Crank out the first few terms: .$\displaystyle 1,\;6,\;13,\;22,\;33,\;46,\;61,\;\hdots$


    Take the difference of consecutive terms,
    . . then the differences of the differences, and so on.

    . . $\displaystyle \begin{array}{cccccccccccc}
    \text{Sequence:} & 1 && 6 && 13 && 22 && 33 && 46 \\
    \text{1stn diff:} && 5 && 7 && 9 && 11 && 13 \\
    \text{2nd diff:} &&& 2 && 2 && 2 && 2 && 2 \end{array}$


    Since the second differences are constant,
    . . the generating function is of the second degree, a quadratic.

    The general quadratic function is: .$\displaystyle f(n) \:=\:an^2 + bn + c$


    Use the first three values of the function and substitute:

    . . $\displaystyle \begin{array}{ccccc}
    f(3) = 13: & 9a + 3b + c &=& 13 & [1] \\
    f(2) = 6: & 4a + 2b + c &=& 6 & [2] \\
    f(1) = 1: & a + b + c & = & 1 & [3] \end{array}$

    . . $\displaystyle \begin{array}{ccccc}
    \text{Subtract [1] - [2]:} & 5a + b &=& 7 & [4] \\
    \text{Subtract [2] - [3]:} & 3a + b &=& 5 & [5] \end{array}$

    . . $\displaystyle \text{Subtract [4] - [5]:} \;\;2a \:=\:2 \quad\Rightarrow\quad \boxed{a \:=\:1}$

    Substitute into [5]: .$\displaystyle 3(1) + b \:=\:5 \quad\Rightarrow\quad\boxed{ b \:=\:2}$

    Substitute into [3]: .$\displaystyle 1 + 2 + c \:=\:1 \quad\Rightarrow\quad\boxed{ c \:=\:-2}$


    Therefore, the generating function is: .$\displaystyle f(n) \;=\;n^2 + 2n - 2$

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: May 26th 2011, 02:33 AM
  2. General term
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 10th 2010, 11:21 PM
  3. general term
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Sep 15th 2009, 03:00 AM
  4. general term
    Posted in the Algebra Forum
    Replies: 5
    Last Post: Sep 2nd 2009, 07:11 AM
  5. Replies: 7
    Last Post: Aug 31st 2007, 08:18 PM

Search Tags


/mathhelpforum @mathhelpforum