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B Algebra question -- Solving 2 simultaneous equations...

  1. Apr 17, 2016 #1
    It is a bit of a long question about series, But what is important is this...

    a+ 2c = 3b

    (3b-2)^2 = 2a * (4c-2)

    It is asking for the ratio c to a

    It have tried a lot of ways. I always end up with this. (Note maybe I not noticing a mistake, Can just anyone confirm that this is solvable? )

    a^2 - 4 a c + 4 = 8 c - 4 c^2
     
  2. jcsd
  3. Apr 17, 2016 #2

    Math_QED

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    You can solve this for a! I got the same expression as you. You can write it as a^2 - 4ac + 4c^2 = 8c - 4.
    Now look at the a^2 - 4ac + 4c^2. You should notice something. Solve for a then and you can express c/a
     
  4. Apr 17, 2016 #3
    a = 2 ( +or- sqrt( c-1) + c)

    Which doesn't make sense in my view, Because the answer is 5 : 2 ( c to a). Did you get the value for c:a as a number or with variables?

    Here is the original question:
    2a, 3b , 4c is a Arithmetic progression
    2a, 3b-2, 4c-2 is a geometric sequence

    Find c : a
     
  5. Apr 17, 2016 #4

    Math_QED

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    It might have been useful to include the original question. I don't have time right now. I will look at it tomorrow.
     
  6. Apr 17, 2016 #5
    This must involve some fancy maneuvers, if it is possible at all. I was able to reduce the single equation we get after eliminating ##b## to the forms:

    ##(a-2c)^2 = 8c-4##

    or

    ##(2c-2)^2 = 4ac-a^2##

    I don't see any way to get ##\frac{c}{a}## out of either of these.
     
  7. Apr 17, 2016 #6

    I wonder if you have left some information out of the problem statement because I get two solutions by brute force search:

    ## [a,b,c] = [1/4,1/2,5/8] ##
    ## [a,b,c] = [1,2,5/2] ##

    Both of these give ##c/a = 5/2## and both sets fit the given progressions. Additionally, these sets fit the progressions but yields the wrong ##c/a## ratio:

    ## [a,b,c] = [4,2,1] ##
    ## [a,b,c] = [16,14,13] ##

    These observations lead me to think there is more to the story than was given here. Is there something missing?
     
  8. Apr 18, 2016 #7
    Apparently, When the teacher gave me the question the he missed some variables in the geometric sequences. Sorry for inconvenience
     
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