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Sequences/Series Sigma Question

  1. Nov 20, 2008 #1
    1. The problem statement, all variables and given/known data

    If [tex]\sum_{n=1}^{4}log_{2}x^{n}=80[/tex], determine the value of x.


    2. Relevant equations

    [tex]S_{n} = \frac{a(r^{n}-1)}{r-1}[/tex]

    [tex]S_{n} = \frac{rt_{n}-a}{r-1}[/tex]


    3. The attempt at a solution

    [tex]log_{2}1+log_{2}x^{2}+log_{2}x^{6}+log_{2}x^{12}=80[/tex]?

    [tex]log_{2}x^{20} = 80[/tex]

    [tex]2^{80}=x^{20}[/tex]

    [tex]x=16[/tex]

    Is this right? Or does x have to be + or - 16 or am I just completely wrong?
     
    Last edited: Nov 20, 2008
  2. jcsd
  3. Nov 20, 2008 #2

    Avodyne

    User Avatar
    Science Advisor

    This looks right, assuming that [tex]log_{2}x^{n}[/tex] means [tex]\log_{2}(x^{n})[/tex] and not [tex](\log_{2}x)^n[/tex].
     
  4. Nov 20, 2008 #3
    Ok well I have two more questions, and rather than clutter the board with another post I'll put them here:

    Question 2:
    1. The problem statement, all variables and given/known data
    A new car costs $42 000 and depreciates 20% the first year, then 15% every year after. What is the car worth in 10 years?

    3. The attempt at a solution
    42 000 + 42 000(0.8) + 33600(0.85) + 33600(0.85)2...

    I'm not sure if this is right so far, but I'm pretty sure I can separate the 42 000(0.8) and the 33600(0.85) right? What should I do?


    Question 3:
    1. The problem statement, all variables and given/known data
    In a geometric sequence, t2+t3=60 while t4+t5=1500. Find the first 3 terms.

    3. The attempt at a solution
    Not really sure how to start this one, I think that t4 is equal to 60r and t5 is equal to 60r2, so can I go:

    60r + 60r2=1500 and solve for r?

    If I do this I end up getting [tex]\frac{-1\pm\sqrt{101}}{2}[/tex] which I'm sure can't be correct.
     
    Last edited: Nov 20, 2008
  5. Nov 20, 2008 #4
    Well, I don't think your first problem is a series. I think you are making it too hard on yourself. It starts at $42,000, loses 20%, and becomes $33600 after one year. I don't think you just keep adding them up.
     
  6. Nov 20, 2008 #5

    Avodyne

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    Science Advisor

    If you're supposed to add them up, then I want that car!:smile:

    As for #3, for a geometric series, [tex]t_n = a\,r^n[/tex] for some constants a and r. So you have two equations in two unknowns.
     
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