ANSWER
The explicit rule is,
[tex]a_n=8({ \frac{3}{4} })^{n - 1} [/tex]
EXPLANATION
The recursive rule for the sequence is given as,
[tex]a_n= \frac{3}{4} a_{n-1}[/tex]
Where,
[tex]a_1=8
[/tex]
Let us find the next term so that we can use it to find the common ratio.
We put
[tex]n = 2[/tex]
into the formula to obtain,
[tex]a_2= \frac{3}{4} a_{2-1}[/tex]
This implies that,
[tex]a_2= \frac{3}{4} a_{1}[/tex]
This will give us
[tex]a_2= \frac{3}{4} (8)[/tex]
[tex]a_2= \frac{3}{1} (2)[/tex]
[tex]a_2= 3 \times 2[/tex]
[tex]a_2= 6[/tex]
The common ratio is
[tex]r = \frac{a_2}{a_1} [/tex]
[tex]r = \frac{6}{8} [/tex]
This reduces to
[tex]r = \frac{3}{4} [/tex]
The explicit rule of the sequence is given by
[tex]a_n=a_1 {r}^{n - 1} [/tex]
We substitute the values to obtain,
[tex]a_n=8({ \frac{3}{4} })^{n - 1} [/tex]
