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Arithmetic/Geometric Progression Problem - Help asap

  1. Aug 15, 2007 #1
    Arithmetic/Geometric Progression Problem SOLVED

    1. The problem statement, all variables and given/known data
    The Three terms a, 1, b are in arithmetic progression. The three terms 1, a, b are in geometric progression. Find the value of a and of b given that a cannot equal b.
    2. Relevant equations
    Arithmetic:
    Here, Un is the nth term.U1 is the first term.
    Un=U1+(n-1)d
    Here, Sn is the Sum of n terms.
    Sn=n/2(2U1+(n-1)d)
    Sn=n/2(U1+Un)
    Geometric
    Un=U1 x r^n-1
    Below, r can't equal 1.
    Sn=U1(r^n-1)/r-1
    Sn=U1(1-r^n)/1-r
    Below, |r|<1
    S=U1/1-r
    3. The attempt at a solution
    Well, lets establish a couple basic things.
    Arithmetic
    U1=a
    U2=1
    U3=b
    U2-U1=d
    U3-U2=d

    U2-U1=U3-U2

    So:
    1-a=b-1
    a+b=2
    a=2-b
    b=2-a
    Then, using the formulas:
    Un=U1 + (n-1)d
    Sn=n/2(2U1 + (n-1)(d))=n/2(U1+Un)

    We can find that:
    a1+a2+a3=s3
    Which is:
    a1+a1+d+a1+2d
    Which is:
    3a+3d=s3

    This is what I have so far, Ill update if I get more.. Im kind of stuck on this and another problem, school starts in 5 hours..(which assumes I won't sleep) so help asap would be appreciated..Thanks!

    Now, I went on to the geometric side, where logic, one of my true strongs, hit me with a reasonable answer.
    If r=u2/u1 and r=u3/u2
    a/1=b/a
    a=b/a
    The only possible and logical thing that popped in my head was, wel A has to be negative.
    So, I plot in -1, but then notice, in arithmetic sequences, b would end up as 1 aswell, which is obviously impossible for an arith.s.
    Therefore, -2 seemed fit. I plot it, everything seems fine, my common difference would be 3, so my next term should be 4.
    Lets plot that into a couple formulas.
    a=b/a
    -2=4/-2. YES
    a+b=2
    -2+4=2 YES

    I revised every one, it does work. Sometimes, logic works better for me than anything, I guess that's my way of solving problems.

    SOLVED
     
    Last edited: Aug 15, 2007
  2. jcsd
  3. Aug 15, 2007 #2

    cristo

    User Avatar
    Staff Emeritus
    Science Advisor

    There is, of course, a way to do it without guessing an answer. From the arithmetic series, you obtain the equation b=2-a, and from the geometric, a=b/a, or, b=a^2. Hence, eliminating b yields a^2+a-2=0, which can be solved to give (a+2)(a-1)=0, which gives the result.
     
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