We know that the rule for a geometric sequence is given by
[tex]a_n=a_1\cdot r^{n-1}[/tex]where a_1 is the first term and a_n is the nth term. From the given information, we also know that
[tex]\begin{gathered} n=2\Rightarrow a_2=24 \\ \text{and} \\ n=4\Rightarrow a_4=384 \end{gathered}[/tex]By substituting these values into the geometric sequence formula, we have
[tex]\begin{gathered} 24=a_1\cdot r^{2-1} \\ \text{which gives} \\ a_1\cdot r=24 \end{gathered}[/tex]and
[tex]\begin{gathered} 384=a_1\cdot r^{4-1} \\ \text{which gives} \\ a_1\cdot r^3=384 \end{gathered}[/tex]So, we have 2 equations:
[tex]\begin{gathered} a_1\cdot r=24\ldots(i) \\ a_1\cdot r^3=384\ldots(ii) \end{gathered}[/tex]We can express a_1 in term of r by dividing equation (i) by r, that is,
[tex]a_1=\frac{24}{r}\ldots(iii)[/tex]By substituting this result into equation (ii), we have
[tex]\frac{24}{r}\cdot r^3=384[/tex]or equivalently,
[tex]24\cdot r^2=384[/tex]Then by dividing both sides by 24, we have
[tex]\begin{gathered} r^2=\frac{384}{24} \\ r^2=16 \end{gathered}[/tex]then
[tex]\begin{gathered} r=\sqrt[]{16} \\ r=4 \end{gathered}[/tex]Once we know the result for r, we can substitute its value into equation (iii) and get
[tex]a_1=\frac{24}{4}=6[/tex]Therefore, the sequence that represents the given values is
[tex]a_n=6\cdot(4)^{n-1}[/tex]In summary, the answers are:
What is the sequence generator? Answer: The generator is the common ratio r. So, r=4.
Write an equation to represent the sequence. Answer: From our last resul, the equation is
[tex]a_n=6\cdot(4)^{n-1}[/tex]