Write an explicit rule in function form for the sequence represented by the given terms. a. f(5)=3 and f(7)=147 b.

Question

Write an explicit rule in function form for the sequence represented by the given terms.

a. f(5)=3 and f(7)=147

b. f(3)=10 and f(5) =1440

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Nevaeh 17 hours 2021-09-15T08:48:32+00:00 1 Answer 0

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    2021-09-15T08:49:36+00:00

    Answer:

    The general relation about a geoemtric sequence is

    a_{n}=a_{1}r^{n-1}

    For the first sequence, we have

    3=a_{1}r^{5-1}

    147=a_{1}r^{7-1}

    Which is a system of equations. We can isolate a variable in the first equation to replace that expression into the second equation.

    a_{1}=\frac{3}{r^{4} }

    147=\frac{3}{r^{4}  }r^{6}

    r^{2}=\frac{147}{3}\\  r=\sqrt[2]{49}\\r=7

    Now, we replace this value to find the other one

    3=a_{1}(7)^{4}\\ a_{1}=\frac{3}{2401}

    Therefore, the explicit rule function is

    a_{n}=\frac{3}{2401}  \times (7)^{n-1}

    Now, we use the same process for the second sequence.

    10=a_{1}r^{3-1}  \\a_{1}=\frac{10}{r^{2} }

    The second equation is

    1440=a_{1}r^{5-1}\\a_{1}=\frac{1440}{r^{4} }

    Now, we solve the following expression

    \frac{10}{r^{2} }=\frac{1440}{r^{4} }

    We solve for r

    \frac{r^{4} }{r^{2} }=\frac{1440}{10}\\r^{2}=144\\ r=\sqrt{144} \\r=12

    Then

    a_{1}=\frac{10}{(12)^{2} }  =\frac{10}{144}=\frac{5}{72}

    Therefore, the function that models the second sequence is

    a_{n}=\frac{5}{72} \times (12)^{n-1}

    Notice that a_{n} is the dependent variable and n is the independent variable.

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45:7+7-4:2-5:5*4+35:2 =? ( )