Let's break down the given information step-by-step to derive the explicit formula for [tex]\( f(x) \)[/tex].
1. Understanding the Recursive Formula:
The recursive formula given is:
[tex]\[
f(x+1) = \frac{2}{3} \cdot f(x)
\][/tex]
This indicates that each term in the sequence is obtained by multiplying the previous term by [tex]\(\frac{2}{3}\)[/tex].
2. Identifying the Sequence Type:
The recursive relationship [tex]\( f(x+1) = \frac{2}{3} \cdot f(x) \)[/tex] shows that this is a geometric sequence. In a geometric sequence, each term is a constant multiple (common ratio [tex]\( r \)[/tex]) of the previous term.
3. General Form of a Geometric Sequence:
The general explicit form of a geometric sequence can be written as:
[tex]\[
f(x) = f(1) \cdot r^{x-1}
\][/tex]
where [tex]\( f(1) \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
4. Identifying [tex]\( f(1) \)[/tex] and [tex]\( r \)[/tex]:
From the problem, we have:
[tex]\[
f(1) = 18
\][/tex]
The common ratio [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{2}{3}
\][/tex]
5. Substitute [tex]\( f(1) \)[/tex] and [tex]\( r \)[/tex] into the General Form:
Plugging [tex]\( f(1) = 18 \)[/tex] and [tex]\( r = \frac{2}{3} \)[/tex] into the explicit formula for a geometric sequence, we get:
[tex]\[
f(x) = 18 \cdot \left( \frac{2}{3} \right)^{x-1}
\][/tex]
Thus, the explicit formula that models the given sequence is:
[tex]\[
f(x) = 18 \left( \frac{2}{3} \right)^{x-1}
\][/tex]
Therefore, the correct option is:
[tex]\[
f(x) = 18 \left( \frac{2}{3} \right)^{x-1}
\][/tex]
