Answer:
The explicit rule for this sequence is; [tex]a_n = 8 \cdot(\frac{3}{4})^{n-1}[/tex]
Step-by-step explanation:
Given the statement: A recursive rule for a geometric sequence is
[tex]a_1=8[/tex] and [tex]a_n =\frac{3}{4}a_{n-1}[/tex]
for n= 2;
[tex]a_2= \frac{3}{4}a_{1}= \frac{3}{4} \cdot 8 = 6[/tex]
Similarly for n = 3;
[tex]a_3 = \frac{3}{4}a_{2}= \frac{3}{4} \cdot 6 = \frac{9}{2}[/tex]
Therefore, we get a geometric sequence i.e,
[tex]8 , 6 , \frac{9}{2}, .......[/tex]
Now, to find the explicit rule for this geometric sequence:
A geometric sequence states that a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio(r).
It is given by: [tex]a_n = a_1r^{n-1}[/tex] where [tex]a_1[/tex] is the first term, r s the common ratio and n is the number of terms;
In the given sequence: [tex]a_1 = 8[/tex], [tex]r = \frac{3}{4}[/tex]
then, the explicit rule for this sequence is;
[tex]a_n = 8 \cdot(\frac{3}{4})^{n-1}[/tex]
