To model the population of the endangered species using a geometric sequence, we start with the provided values:
- The initial population in 2018 (n=1) is 240.
- The subsequent populations are 360, 540, and 810.
We observe that this is indeed a geometric sequence, and we need to find the common ratio.
- From 240 to 360: \( \frac{360}{240} = 1.5 \)
- From 360 to 540: \( \frac{540}{360} = 1.5 \)
- From 540 to 810: \( \frac{810}{540} = 1.5 \)
Thus, the common ratio is \( r = 1.5 \).
The formula for the nth term of a geometric sequence can be expressed as:
\[ f(n) = a \cdot r^{(n-1)} \]
where:
- \( a \) is the first term, which is 240.
- \( r \) is the common ratio, which is \( \frac{3}{2} \) or 1.5.
- \( n \) is the term number.
We can now put these into our formula:
\[ f(n) = 240 \left( \frac{3}{2} \right)^{(n-1)} \]
Thus, the correct function that models this situation is:
f(n) = 240(32)^(n−1)
However, this expression looks incorrect. The correct function actually is:
f(n) = 240(3/2)^(n−1)
Since none of the options provided match this format perfectly, you should select the one that is closest in value to representative of the common ratio \(1.5\) which is indicated in the options you've included.
Based on your options, it should be:
f(n)=240(32)n−1 could mathematically represent the errors but typically refers back to the fundamental ratio of population growth that should be \( f(n) = 240 \left( \frac{3}{2} \right)^{(n-1)} \)
Please recheck your given options carefully.