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Math Help - APGP Combined

  1. #1
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    APGP Combined

    First 3 terms of a GP are 2nd, 10th and 26th terms of an AP respectively. Show that common ratio of GP is 2.

    How the heck do you do that? Thanks for helping!
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  2. #2
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    Quote Originally Posted by zeromeyzl View Post
    First 3 terms of a GP are 2nd, 10th and 26th terms of an AP respectively. Show that common ratio of GP is 2.

    How the heck do you do that? Thanks for helping!
    Start by considering the following three equations:

    a = b + d .... (1)

    ar = b + 9d .... (2)

    ar^2 = b + 25d .... (3)
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  3. #3
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    Hello, zeromeyzl!

    You're expected to know the basics for an AP and GP.

    The n^{th} term of an AP is: . a_n \:=\:a + (n-1)d\quad\begin{Bmatrix}a = \text{first term} \\ d = \text{common difference}\end{Bmatrix}


    The n^{th} term of a GP is: . a_n \:=\:ar^{n-1} \quad \begin{Bmatrix}a = \text{first term} \\ r = \text{common ratio} \end{Bmatrix}


    The first 3 terms of a GP are 2nd, 10th and 26th terms of an AP respectively.
    Show that common ratio of GP is 2.
    For the AP: . \begin{array}{ccc}a_2&=& a + d \\ a_{10} &=& a + 9d \\ a_{26} &=& a + 25d \end{array}

    Since these are in GP: . \begin{array}{cccccccc}(a+d)r &=& a+9d & \Rightarrow & r &=& \dfrac{a+9d}{a+d} & {color{blue}[1]}\\ \\[-3mm]<br />
(a+9d)r &=& a + 25d & \Rightarrow & r &=& \dfrac{a+25d}{a+9d} & {\color{blue}[2]} \end{array}

    Equate [1] and [2]: . \frac{a+9d}{a+d} \:=\:\frac{a+25d}{a+9d} \quad\Rightarrow\quad (a+9d)^2 \:=\:(a+d)(a+25d)

    . . a^2 + 18ad + 81d^2 \:=\:a^2 + 26ad + 25d^2 \quad\Rightarrow\quad 56d^2 \:=\:8ad


    Since d \ne 0, we can divide by 8d\!:\;\;7d \:=\:a


    Substitute into [1]: . r \;=\;\frac{7d+9d}{7d+d} \:=\:\frac{16d}{8d} \quad\Rightarrow\quad\boxed{ r \:=\:2}

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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    Start by considering the following three equations:

    a = b + d .... (1)

    ar = b + 9d .... (2)

    ar^2 = b + 25d .... (3)
    It follows that:

    r = \frac{b + 9d}{b+d}

     \Rightarrow r(b + d) = b + 9d

     \Rightarrow rb + rd = b + 9d \Rightarrow b(r-1) = d(9 - r)

    \Rightarrow \frac{b}{d} = \frac{9-r}{r-1} .... (A)


    r = \frac{b + 25d}{b+9d} \Rightarrow \frac{b}{d} = \frac{\text{Left for you}}{r-1} .... (B)


    Therefore ....
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