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Math Help - image Geometric Sequence and Recursive Definition

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    image Geometric Sequence and Recursive Definition

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    Last edited by SarahGr; May 14th 2009 at 07:28 PM.
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    Hello, SarahGr!

    1) In a Geometric Sequence: . t_3 = 4\:\text{ and }\:t_6= \frac{4}{27} . . Find t_{10}
    We have: . \begin{array}{cccc}<br />
t_3 = 4 & \Rightarrow & t_{_1}r^2 \:=\:4 & [1] \\<br />
t_6 = \frac{4}{27} & \Rightarrow & t_{_1}r^5 \:=\:\frac{4}{27} & [2] \end{array}

    Divide [2] by [1]: . \frac{t_{_1}r^5}{t_{_1}r^2} \:=\:\frac{\frac{4}{27}}{4} \quad\Rightarrow\quad r^3 \:=\:\frac{1}{27} \quad\Rightarrow\quad r \:=\:\frac{1}{3}

    Substitute into [1]: . t_{_1}\left(\tfrac{1}{3}\right)^2 \:=\:4 \quad\Rightarrow\quad t_{_1} \:=\:36


    Therefore: . t_{10} \;=\;t_{_1}r^9 \;=\;36\left(\tfrac{1}{3}\right)^9 \;=\;\frac{36}{19,\!683} \;=\;\frac{4}{2187}




    2) Give a recursive definition for the sequence: . 1, 4, 13, 40, \hdots
    Take the differences of consecutive terms . . .

    . . \begin{array}{ccccccccc}\text{Sequence} & 1 && 4 && 13 && 40 \\<br />
\text{Difference} & & 3 & & 9 & & 27 \end{array}


    Each term is the preceding term plus a power of 3.

    . . . . . a_n \;=\;a_{n-1} + 3^{n-1}

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    Last edited by SarahGr; May 14th 2009 at 07:28 PM.
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