To model the population of the endangered species with a geometric sequence, we start by identifying the first term and the common ratio.
The initial population in 2018 is given as:
- \( P(1) = 240 \) (for n = 1)
- \( P(2) = 360 \) (for n = 2)
- \( P(3) = 540 \) (for n = 3)
- \( P(4) = 810 \) (for n = 4)
To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{P(2)}{P(1)} = \frac{360}{240} = 1.5 \]
Now we can express the population function in general terms. The first term \( a = 240 \) and the common ratio \( r = 1.5 \). The formula for the \( n \)-th term of a geometric sequence is given by:
\[ f(n) = a \cdot r^{n-1} \]
Substituting in the known values:
\[ f(n) = 240 \cdot (1.5)^{n-1} \]
Now, looking at the available responses, the function that matches our expression is:
\[ f(n) = 240 \left( \frac{3}{2} \right)^{n-1} \]
Therefore, the correct answer is:
f(n) = 240(3/2)^(n-1)