Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sequences Series geometric series or an arithmetic series?

  1. Apr 8, 2007 #1
    This is the sequence: 1, 2, 5, 14, 41, 122

    1. Is this a geometric series or an arithmetic series?
    2. I know the formula is a sub n=[3^(n-1)+1]/2, but how do you get that from a sub n=a sub 1 * r^(n-1), which is the geometric formula for series.
    Last edited: Apr 8, 2007
  2. jcsd
  3. Apr 8, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    1. Neither.

    2. I assume your sequence should be 1,2,5,14,41,122

    You don't get it from a geometric series, since it isn't a geometric sequence.
  4. Apr 8, 2007 #3


    User Avatar
    Science Advisor
    Gold Member

    It is neither - geometric series means constant ratio between terms, arithmetic means constant difference.
  5. Apr 8, 2007 #4
    So how would you get it then, with what formula or method?
  6. Apr 8, 2007 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Well, if you are clever, you will see that if n>=2, then we have:


    Assuming this is the pattern for all the next numbers, you may derive that explicit formula.
  7. Apr 8, 2007 #6
    So there is no definite way of determining that formula? I see that to get from a term to another you add first 1, then 3, then 9, then 27, then 81 which are multiples of three. Does this have anything to do with determining the formula?

  8. Apr 8, 2007 #7


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    A finite sequence can be extended in infinitely many ways, i.e, there exists an infinity of patterns to choose from.
  9. Apr 8, 2007 #8
    A pattern that fits this is: [tex]F(N)=\frac{3^N+1}{2}, N=0,1,2..[/tex]
  10. Apr 9, 2007 #9
    I know that that is the pattern, but I was just wondering how to figure that out with a formula or something.
  11. Apr 9, 2007 #10

    2) you can convert it to geometric series >>

    t=1+2+5+14+41+122..... Tn ------(1)
    t= 1+2+5+14+41+122....Tn-Tn-1+Tn ------(2)

    now eq 1- 2 and you'll get geometric series .
  12. Apr 9, 2007 #11
    Can you please explain that more?
  13. Apr 9, 2007 #12

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Post 5 gives you a difference equation to solve. You also have the solution as a closed formula, to the difference equation, so it is a simple induction argument to verify it.
  14. Apr 9, 2007 #13
    And yet, in the context of a math education, only one of these is considered to have a "formula". Although we with our modern sensibilities abhor this notion of formula, Euler would concur.
  15. Apr 10, 2007 #14

    Tn=1+2+5+14+41+122..... Tn ------(1)
    Tn=0+1+2+5+14+41+122....Tn-1+Tn ------(2)

    -------------------------------------------------------------- Eq 1- 2

    0 = 1+1+3+3^2+3^3+3^4+................ (Tn-Tn-1) - Tn

    now transfer that Tn to that side (where zero is) & other side will have (n-1) terms and if you'll not include 1 of (first one) of series then terms will be (n-2) .that fact is that when you get such type of serieses you have to see for diffrences of series .

    Last edited: Apr 10, 2007
  16. Apr 12, 2007 #15

    Usualy, assuming that we deal with a homogeneous linear sequence, the recurrence relation which we have to seek is that having the smallest degree, in this case a[k]-4a[k-1]+3a[k-2]=0 which gives immediately the closed form from the OP (k=3,4...).

    [The characteristic equation is r^2-4*r+3=0 ---> r1=1; r2=3

    Therefore we must seek a solution of the form a[k]=A*(1)^k+B*[(3)^k] (1); A,B = constants

    we have a[k=3]=5 and a[k=4]=14 ---> replacing k in (1) with 3 and 4 results a system of equations from which A=1/2 and B=1/6.]
    Last edited: Apr 12, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Sequences Series geometric series or an arithmetic series?
  1. Arithmetic series (Replies: 4)

  2. Geometric Series (Replies: 2)