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Math Help - Geometric Series issue.

  1. #1
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    Geometric Series issue.

    Having a bit of trouble with the following problem:

    The sum of the first three terms of a geometric series is 21. The sum of terms 4 through 6 is 168. Find the first three terms.

    Would greatly appreciate help.
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  2. #2
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    Re: Geometric Series issue.

    Hello, Ventus!

    Do you know anything about Geometric Series?


    The sum of the first three terms of a geometric series is 21.
    The sum of terms 4 through 6 is 168.
    Find the first three terms.

    The first six terms are: . a,\; ar,\; ar^2,\; ar^3,\; ar^4,\; ar^5

    We are told: . \begin{Bmatrix}a + ar + ar^2 \:=\: 21 & \Rightarrow & a(1+r + r^2) \:=\: 21 & [1] \\ \\[-3mm] ar^3 + ar^4 + ar^5 \:=\: 168 & \Rightarrow & ar^3(1+r+r^2) \:=\: 168 & [2] \end{Bmatrix}

    Divide [2] by [1]: . \frac{ar^3(1+r+r^2)}{a(1+r+r^2)} \:=\:\frac{168}{21}

    . . And we have: . r^3 \:=\:8 \quad\Rightarrow\quad \boxed{r \:=\:2}

    Substitute into [1]: . a(1+2+2^2) \:=\:21 \quad\Rightarrow\quad 7a \:=\:21 \quad\Rightarrow\quad \boxed{a \:=\:3}


    Therefore, the first three terms are: . (a,\:ar,\:ar^2) \;=\;(3,\:6,\:12)

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  3. #3
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    Re: Geometric Series issue.

    Quote Originally Posted by Soroban View Post
    Hello, Ventus!

    Do you know anything about Geometric Series?



    The first six terms are: . a,\; ar,\; ar^2,\; ar^3,\; ar^4,\; ar^5

    We are told: . \begin{Bmatrix}a + ar + ar^2 \:=\: 21 & \Rightarrow & a(1+r + r^2) \:=\: 21 & [1] \\ \\[-3mm] ar^3 + ar^4 + ar^5 \:=\: 168 & \Rightarrow & ar^3(1+r+r^2) \:=\: 168 & [2] \end{Bmatrix}

    Divide [2] by [1]: . \frac{ar^3(1+r+r^2)}{a(1+r+r^2)} \:=\:\frac{168}{21}

    . . And we have: . r^3 \:=\:8 \quad\Rightarrow\quad \boxed{r \:=\:2}

    Substitute into [1]: . a(1+2+2^2) \:=\:21 \quad\Rightarrow\quad 7a \:=\:21 \quad\Rightarrow\quad \boxed{a \:=\:3}


    Therefore, the first three terms are: . (a,\:ar,\:ar^2) \;=\;(3,\:6,\:12)

    Oh I see, I neglected to factor out a and ar^3.

    Thank you very much! I'll be going into my test with much added confidence.
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