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Math Help - Geomtric sequence question

  1. #1
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    Geomtric sequence question

    Hey guys just started a new subject in maths today on geometric sequence

    Any way here is the question I am having trouble with..


    Find the value of x such that 3 - x; x; 2 - x are successive terms of a G.S and state the value of common ratio, r.
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  2. #2
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    If 3-x\, ,x\, ,2-x are in GP, then

    x^2 = (3-x)(2-x)
    \Rightarrow x^2 = 6 - 5x + x^2
    \Rightarrow x = \frac65

    So the common ratio is \frac{x}{3-x} = \frac{\frac65}{3-\frac65} = \frac{6}{15-6} = \frac23
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  3. #3
    Member Danshader's Avatar
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    well geometric progression has the form of:
     ar^n


    the first term:[tex] ar^0 = a
    the second term:[tex] ar^1 = ar
    the third term:  ar^2 = ar^2

    to obtain r we divide a term with its previous term,

     \frac{ar^n}{ar^{n-1}} = r

    e.g.
     \frac{2ndterm}{1stterm} = \frac {ar}{a}

    and this is true through out the progression
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  4. #4
    Super Member wingless's Avatar
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    a_n = ..., ~3-x , ~x , ~2 - x,~...

    If this is a geometric sequence, every new value will be (previous . r).

    a_{n+1} = r\cdot a_n

    So, 2-x = x \cdot r

    x = (3-x)\cdot r

    Then, we can write r = \frac{2-x}{x} = \frac{x}{3-x}

    Now just solve it. Is it OK so far? This is the same as what isomorphism did, I only wanted to explain because I saw that you've just started this topic.


    Now let's find a rule for these kinds of questions.

    If a is a term of a geometric sequence, the next terms will be a\cdot r and a \cdot r^2.

    a,~a.r, ~a.r^2

    You can easily see that (a)\cdot (a.r^2) = (a.r)^2.

    So, a_{n-1} \cdot a_{n+1} = a_n^2
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  5. #5
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    Hello, smplease!

    Find the value of x such that 3 - x,\; x,\; 2 - x are successive terms of a G.S
    State the value of common ratio, r.
    From the definition of the common ratio, we have: . \begin{array}{cccc}\dfrac{x}{3-x} &=& r & {\color{blue}[1]} \\ \\ [-3mm] \dfrac{2-x}{x} &=& r & {\color{blue}[2]} \end{array}

    Equate [1] and [2]: . \frac{x}{3-x} \:=\:\frac{2-x}{x}\quad\Rightarrow\quad\boxed{ x \:=\:\frac{6}{5}}

    Substitute into [2]: . r \:=\:\frac{2-\frac{6}{5}}{\frac{6}{5}}\quad\Rightarrow\quad\box  ed{ r \:=\:\frac{2}{3}}

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