Respuesta :
Answer:
The general relation about a geoemtric sequence is
[tex]a_{n}=a_{1}r^{n-1}[/tex]
For the first sequence, we have
[tex]3=a_{1}r^{5-1}[/tex]
[tex]147=a_{1}r^{7-1}[/tex]
Which is a system of equations. We can isolate a variable in the first equation to replace that expression into the second equation.
[tex]a_{1}=\frac{3}{r^{4} }[/tex]
[tex]147=\frac{3}{r^{4} }r^{6}[/tex]
[tex]r^{2}=\frac{147}{3}\\ r=\sqrt[2]{49}\\r=7[/tex]
Now, we replace this value to find the other one
[tex]3=a_{1}(7)^{4}\\ a_{1}=\frac{3}{2401}[/tex]
Therefore, the explicit rule function is
[tex]a_{n}=\frac{3}{2401} \times (7)^{n-1}[/tex]
Now, we use the same process for the second sequence.
[tex]10=a_{1}r^{3-1} \\a_{1}=\frac{10}{r^{2} }[/tex]
The second equation is
[tex]1440=a_{1}r^{5-1}\\a_{1}=\frac{1440}{r^{4} }[/tex]
Now, we solve the following expression
[tex]\frac{10}{r^{2} }=\frac{1440}{r^{4} }[/tex]
We solve for [tex]r[/tex]
[tex]\frac{r^{4} }{r^{2} }=\frac{1440}{10}\\r^{2}=144\\ r=\sqrt{144} \\r=12[/tex]
Then
[tex]a_{1}=\frac{10}{(12)^{2} } =\frac{10}{144}=\frac{5}{72}[/tex]
Therefore, the function that models the second sequence is
[tex]a_{n}=\frac{5}{72} \times (12)^{n-1}[/tex]
Notice that [tex]a_{n}[/tex] is the dependent variable and [tex]n[/tex] is the independent variable.
