To model the population of the endangered species using a geometric sequence, we start by identifying the first term and the common ratio.
The given population values are:
- Year 0 (2018): \(240\)
- Year 1 (2019): \(360\)
- Year 2 (2020): \(540\)
- Year 3 (2021): \(810\)
We can see that each term can be obtained from the previous term by multiplying by a common ratio.
To find the common ratio (r), we can divide the second term by the first term:
\[ r = \frac{360}{240} = 1.5 \]
Also, check the ratio between the second and third terms:
\[ r = \frac{540}{360} = 1.5 \]
And the ratio between the third and fourth terms:
\[ r = \frac{810}{540} = 1.5 \]
Thus, the common ratio \(r\) is indeed \( \frac{3}{2} \) (or 1.5).
The general formula for a geometric sequence is given by:
\[ f(n) = a \cdot r^{(n-1)} \]
where:
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.
In this case:
- \(a = 240\)
- \(r = \frac{3}{2}\)
Thus, the function that models this situation is:
\[ f(n) = 240 \left( \frac{3}{2} \right)^{(n-1)} \]
Among the provided options, the correct response is:
f(n) = 240(32)^{n-1}
This option seems to contain a typographical error for the common ratio; it should correctly indicate \( \frac{3}{2} \) instead of showing \( 32 \).
Hence, the modeled function should be confirmed as:
f(n) = 240(1.5)^{(n-1)} or f(n) = 240\left(\frac{3}{2}\right)^{(n-1)} (which is the mathematically accurate statement of the sequence).
As none of the provided options are exactly correct, you can clarify or correct them based on the correct parameters discussed.