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Answer :
We are given the recurrence relation
[tex]$$
f(x+1)=\frac{2}{3}f(x)
$$[/tex]
with the initial condition
[tex]$$
f(0)=108.
$$[/tex]
This recurrence relation defines a geometric sequence. The general form of a geometric sequence is
[tex]$$
f(n)=f(0) \cdot r^n,
$$[/tex]
where [tex]$r$[/tex] is the common ratio. Here, the common ratio is
[tex]$$
r=\frac{2}{3}.
$$[/tex]
Thus, the explicit formula for the sequence is
[tex]$$
f(n)=108\left(\frac{2}{3}\right)^n.
$$[/tex]
Let's calculate a few terms to understand the behavior of the function:
1. When [tex]$n=0$[/tex]:
[tex]$$
f(0)=108\left(\frac{2}{3}\right)^0=108\cdot 1=108.
$$[/tex]
2. When [tex]$n=1$[/tex]:
[tex]$$
f(1)=108\left(\frac{2}{3}\right)^1=108\cdot\frac{2}{3}=72.
$$[/tex]
3. When [tex]$n=2$[/tex]:
[tex]$$
f(2)=108\left(\frac{2}{3}\right)^2=108\cdot\frac{4}{9}=48.
$$[/tex]
4. When [tex]$n=3$[/tex]:
[tex]$$
f(3)=108\left(\frac{2}{3}\right)^3=108\cdot\frac{8}{27}\approx32.
$$[/tex]
Notice that each successive term is smaller than the previous one because the common ratio [tex]$\frac{2}{3}$[/tex] is less than 1. This confirms that the sequence is decreasing exponentially.
When choosing the appropriate graph from the available options (labeled with numbers such as 150, 135, and 120), we need the graph that shows this exponential decay starting at [tex]$108$[/tex] and then decreasing (with values like [tex]$108$[/tex], [tex]$72$[/tex], [tex]$48$[/tex], [tex]$32$[/tex], …).
Based on the pattern of the computed terms, the graph that correctly represents [tex]$f(n)=108\left(\frac{2}{3}\right)^n$[/tex] exhibits an exponential decay. Therefore, the correct graph is identified as option 3.
The final answer is: [tex]$\boxed{3}$[/tex].
[tex]$$
f(x+1)=\frac{2}{3}f(x)
$$[/tex]
with the initial condition
[tex]$$
f(0)=108.
$$[/tex]
This recurrence relation defines a geometric sequence. The general form of a geometric sequence is
[tex]$$
f(n)=f(0) \cdot r^n,
$$[/tex]
where [tex]$r$[/tex] is the common ratio. Here, the common ratio is
[tex]$$
r=\frac{2}{3}.
$$[/tex]
Thus, the explicit formula for the sequence is
[tex]$$
f(n)=108\left(\frac{2}{3}\right)^n.
$$[/tex]
Let's calculate a few terms to understand the behavior of the function:
1. When [tex]$n=0$[/tex]:
[tex]$$
f(0)=108\left(\frac{2}{3}\right)^0=108\cdot 1=108.
$$[/tex]
2. When [tex]$n=1$[/tex]:
[tex]$$
f(1)=108\left(\frac{2}{3}\right)^1=108\cdot\frac{2}{3}=72.
$$[/tex]
3. When [tex]$n=2$[/tex]:
[tex]$$
f(2)=108\left(\frac{2}{3}\right)^2=108\cdot\frac{4}{9}=48.
$$[/tex]
4. When [tex]$n=3$[/tex]:
[tex]$$
f(3)=108\left(\frac{2}{3}\right)^3=108\cdot\frac{8}{27}\approx32.
$$[/tex]
Notice that each successive term is smaller than the previous one because the common ratio [tex]$\frac{2}{3}$[/tex] is less than 1. This confirms that the sequence is decreasing exponentially.
When choosing the appropriate graph from the available options (labeled with numbers such as 150, 135, and 120), we need the graph that shows this exponential decay starting at [tex]$108$[/tex] and then decreasing (with values like [tex]$108$[/tex], [tex]$72$[/tex], [tex]$48$[/tex], [tex]$32$[/tex], …).
Based on the pattern of the computed terms, the graph that correctly represents [tex]$f(n)=108\left(\frac{2}{3}\right)^n$[/tex] exhibits an exponential decay. Therefore, the correct graph is identified as option 3.
The final answer is: [tex]$\boxed{3}$[/tex].
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