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Thread: Geometric Sequences..

  1. #1
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    Geometric Sequences..

    I don't understand how the book got the answer of 5/9

    The question is:

    Given the two terms of the geometric sequence, find the individual term.
    t4=5 t8=405 find t2

    i used the formula tn = ar^n-1 for this

    t8/t4 = 405/5 = ar^7 / ar^3
    =81 = ar^4
    r=3

    5=a(3)^4
    5=81a
    a = 5/81

    t2=(5/81)(3)^1
    =5/27
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  2. #2
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    You should know that $\displaystyle \displaystyle t_n = a\,r^{n - 1}$.

    Therefore $\displaystyle \displaystyle t_4 = a\,r^3$ and $\displaystyle \displaystyle t_8 = a\,r^7$.


    From this we have

    $\displaystyle \displaystyle a\,r^7 = 405$

    $\displaystyle \displaystyle a\,r^3 = 5$.


    Dividing the first equation by the second gives

    $\displaystyle \displaystyle r^4 = 81$

    $\displaystyle \displaystyle r = \pm 3$.


    Substituting back into one of the equations:

    $\displaystyle \displaystyle 405 = a\,(\pm 3)^7$

    $\displaystyle \displaystyle 405 = \pm 2187 a$

    $\displaystyle \displaystyle a = \pm\frac{5}{27}$.


    So there are two possible sequences, the first with $\displaystyle \displaystyle a = -\frac{5}{27}$ and $\displaystyle \displaystyle r = -3$, the second with $\displaystyle \displaystyle a = \frac{5}{27}$ and $\displaystyle \displaystyle r = 3$.

    Either way, you should find that

    $\displaystyle \displaystyle t_2 = a\,r = \frac{5}{27}\cdot 3 = \frac{5}{9}$ or $\displaystyle \displaystyle t_2 = -\frac{5}{27}\left(-3\right) = \frac{5}{9}$.
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