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Number Of Terms In A G.p

  1. Apr 6, 2012 #1
    1. The problem statement, all variables and given/known data
    How many terms
    has the G.p whose
    2nd term is 1/2
    and common ratio
    and the last term
    1/4 and 1/128
    respectiveley?

    From the question, it can be deduced that the common ratio, r =1/4 and the last term = 1/128

    2. Relevant equations
    The Nt term of a G.p =
    arn-1

    3. The attempt at a solution
    Given that r, the
    common ratio=1/4.
    T2, the second
    term is =1/2.
    Using the formula
    arn-1
    where a = first
    term and r is =
    common ratio.
    Thus:T2=a(1/4)2-1
    =1/2
    a(1/4) = 1/2
    a/4 = 1/2
    2a = 4
    a = 4/2
    a = 2
    Recall that Tn of
    G.p = arn-1
    To find the Nt
    term, we need to put a, r and the last term in equation
    arn-1 = 1/128
    2(1/4)n-1 =1/128
    I multiplied through
    by 1/2
    2 * 1/2(1/4)n-1 =
    1/128 * 1/2
    (1/4)n-1 = 1/256
    I don't know how
    to deal with the
    power n-1 in other
    to procceed
    further and arive
    at the answer.
     
    Last edited: Apr 6, 2012
  2. jcsd
  3. Apr 6, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Why not just "do the arithmetic"? What is 22? What is 23? What is 24? ...

    This works because this problem happens to be "set up" so that it is easy- 256 happens to be an integer power of 4. More generally to solve an= b for n, you would have to use logarithms. Taking the logarithm of both sides, log(an)= n log(a)= log(b) so n= log(b)/log(a).

    You can use logarithm to whatever base is convenient.
     
  4. Apr 6, 2012 #3
    Logarithm! Hm! I have never seen where logarithm is aplied in problem relating to G.p such at this; atleast not in my level. Anyway, am not afraid to learn new things; maybe you should start while I watch and learn. No abuse intended, am just expressing my mind.
     
  5. Apr 6, 2012 #4
    2, 1/2, 1/8, 1/32, 1/128 (5 terms)
     
  6. Apr 6, 2012 #5
    Thank you but I must add this:
    the answer is not the problem but how or what procedure did you employ to arrive at that answer; that is my main problem.
     
  7. Apr 6, 2012 #6
    If the second term is 1/2, and the common ratio is 1/4, then the first term must have been (1/2)/(1/4)=2. Once you know this, you just write out the terms in order, and count the number of terms required to reach 1/128.

    Chet
     
  8. Apr 7, 2012 #7
    As I stated earliar, the answer is not the problem but what is the problem is how to do and get that answer. If you watch my working from the begining, you will see that the way I got the value for my first term, a, is quite different from the way you got yours. From your working you have only gotting the value for the first term. If this your working is reliable, can you use it to get the value for the third term, fourth term and finally the fifth term.
     
  9. Apr 7, 2012 #8
    Well, no, that would just be enumerating the series. Using the formula, you can directly infer the starting term and then the position of the given term. Working through the series is all very well in this case, and a great check on your working, but the general idea of getting a particular term in the series without doing that is more useful.

    Which is what you said anyway, Chikis, and asked for help on diong that with 2.(1/4)n-1 = 1/128

    So specifically on that:
    [tex]
    \begin{align}
    2(\frac{1}{4})^{n-1} &= \frac{2}{4^{n-1}}\\
    &= \frac{2}{2^{2n-2}}\\
    &= \frac{1}{2^{2n-3}}
    \end{align} \\

    128 = 2^7 [/tex]
    and solve for n.
     
    Last edited: Apr 7, 2012
  10. Apr 8, 2012 #9
    Well, all these are not necessary, I have already gotten the solution to the problem.
    Here is the solution:
    am starting from where I stopped

    (1/4)n-1 = 1/256
    (4-1)n-1 = (256-1)
    (4)-n+1 = (28)-1
    (22)-n+1 = (2)-8
    2-2n+2 = 2-8
    the two equation are now equal, so we now equate the both indices of both equation and solve for n,
    -2n+2 = -8
    -2n = -8-2
    -2n = -10
    n = -10/-2
    n = 5
    and that is the number of terms in the G.p.
    If we want to still go futher and write out the value of each terms, then we get it by mutiplying the the ratio with each term to get the next term after it:
    we arleady know from the given question and and our calculation that the common ratio, r, the first term, a, the second term, T2 and the last term, T3 are 1/4, 2, 1/2 and 1/128; what is remaining for us now is to get the value for the third term, T3 and the the fourth term, T4 to get it we do the following,
    T3 = 1/2*1/4 = 1/8
    T4 = 1/8*1/4 = 1/32
    :. The complete G.p is as follows
    2, 1/2, 1/8, 1/32, 1/128.
    Thus the problem is solved:-D.
     
  11. Apr 9, 2012 #10
    According to the definition of a geometric progression, each term in the progression is equal to the previous term times the common ratio. That's all the information I used.
    Silly me. I thought you were just looking for a solution to the problem. I didn't realize it had to be done using exactly what you learned in Algebra class. Here is another lesson in algebra:

    (1/4)**(n-1)=(1/256)

    4**(n-1)=256=4**(4)

    n-1=4

    n=5

    Chet
     
  12. Apr 9, 2012 #11
    I don't understand your own working from this line:
    (1/4)**(n-1)=(1/256)

    4**(n-1)=256=4**(4)

    Can you explain please!!
     
  13. Apr 10, 2012 #12
    ([itex]\frac{1}{4}[/itex])(n-1)=[itex]\frac{1}{256}[/itex]

    4(n-1)=256=44

    n-1=4

    n=5
     
  14. Apr 10, 2012 #13
    I don't undestand all these written in bold form:
    ([itex]\frac{1}{4}[/itex])(n-1)=[itex]\frac{1}{256}[/itex]
    Write it in such a way that I may understand it. Don't you see my own? Is there anything you don't understand in my own working? Just put it in a simple form just like mine.
     
  15. Apr 11, 2012 #14
    Yes. To me on my computer, the equations look pretty much the same as the ones in your replies. I used the PF equation editor... I think they call it latex. I don't know what else I can do.

    Chet
     
  16. Apr 11, 2012 #15
    Then how do I get the exact equation of your working?
     
  17. Apr 14, 2012 #16
    I'm going to try to write it out in words. I hope this works for you.

    (1/4) to the power (n-1) is equal to (1/256).

    (1/256) is equal to (1/4) to the power 4.

    So, (1/4) to the power (n-1) is equal to (1/4) to the power 4.

    Therefore, (n-1) = 4

    n=5
     
  18. Apr 14, 2012 #17
    That's a good one. Here is what you mean in figures.
    (1/4)n-1 = (1/256)
    (1/4)n-1 = (1/4)4
    n-1 = 4
    n = 4+1
    n = 5.
    That's is a good one. Atleast one's brain will be sharpened. Thank you for that idea.
     
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