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Summing Geometric Sequences am i doing this right?

  1. Oct 16, 2006 #1
    We learned in class how to find the sum of any geometric sequence with the following formula:

    Let x = Sum Of Geometric Sequence;

    x = [Take mythical next term - real term]/(ratio - 1);

    The real term is the first term of the sequence and the mythical next term would be the next term for example after this sequence:

    ex:
    1+2+2^2+2^3 +...+2^n = ?

    first term is : 1
    mythical next term is: 2^(n+1)
    ratio: 2
    when I say ratio I mean what you multiply each term to get to the next and you can see its 2 here.

    x = [2^(n+1)-1]/(2-1)

    So that would be the formula to sum up the sequence...so that one was easy but this is the one that i don't understand:


    2^n + 2^(n-1) x 3 + 2^(n-2) x 3^2 + 2^(n-3) x 3^3 + .... + 2^3 x 3^(n-3)


    I don't know if i'm getting the ratio right or not...
    I see each term is getting multiplied by 3, the first term is just 3^0 = 1, then 3^1, 3^2...

    I see n is being decremented by 1 each time and its a power of 2, so would that be: 1/2^n

    So the ratio i figured out is: 3/2^n

    Now the mythical next term I got is: 2^4 x 3^(n-4)
    or do i not mess with the 2^3? and leave it as 2^3 x 3^(n-4) ?

    The 1st real term is: 2^n

    So here is the formula i come out with:

    Sum of geometric sequence = [ [2^4 x 3^(n-4)] - 2^n ]/[(3/2^n)-1]

    I'm testing for n = 3, n = 4, and n = 5 to see if itss right...
    for n = 3
    2^3 + 2^2 x 3 + 2 x 3^2 = 38

    now if i plug 3 into the formula:
    [2^4 x 3^(-1) - 2^3 ] / (3/(2^3) - 1) = 64/15 which isn't 38...

    Do you see what i'm doing wrong? I think i'm screwing up on the ratio and the mythical next term, the next term in the sequence, any help would be great. I have to do it the way the professor showed us with that general formula:

    Let x = Sum Of Geometric Sequence;

    x = [Take mythical next term - real term]/(ratio - 1);


    Thanks!
     
  2. jcsd
  3. Oct 16, 2006 #2

    StatusX

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    The ratio is just 3/2, not 3/2^n. Multiply the last term by this ratio to get the "mythical next term."
     
  4. Oct 16, 2006 #3
    Thanks for the reply


    oh, multiplying by the ratio would make sense :bugeye:

    Okay now i have as the last term: 4/9 x 3^n as the next term
    and ratio: 3/2

    x = [ (4/9) x 3^n - 2^n ]/[(3/2) - 1]

    if i plug in n = 3, i get x = 8

    if i go back to the sequence and add up the first 3 terms, after letting n = 3, i get the following:

    2^3 + 2^(2) x 3 + 2 x 3^2 = 38

    Is this not how you check to see if the formula is correct or not? If i let n = 3 in the formula, it should sum the first 3 terms right?
     
  5. Oct 16, 2006 #4

    StatusX

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    No, the power of 2 runs from n down to 3, so for n=3 there is only one term, 2^3=8.
     
  6. Oct 16, 2006 #5
    yep that works! thanks!

    I think i get what your saying, so if I wanted to check for n = 4, i would just add up
    2^4 + 2^(4-1) = 20

    and 5 i would add up the first 3 terms with n = 5 right?
     
  7. Oct 16, 2006 #6

    StatusX

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    Well that sum isn't right: it's missing the 3's, and 2^4+2^3=16+8=24. What you said about the number of terms is right, but why don't you just check the new formula you found to see if it works?
     
  8. Oct 16, 2006 #7
    alroight now i get it! hah sorry i'm coming down from a cafine overdose i think.
     
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