To model the given population data as a geometric sequence, we first note the initial term and the common ratio.
- The initial population in 2018 (when \( n = 1 \)) is \( a_1 = 240 \).
- The subsequent populations are:
- \( a_2 = 360 \)
- \( a_3 = 540 \)
- \( a_4 = 810 \)
To find the common ratio \( r \), we can find the ratio between consecutive terms:
-
From 240 to 360: \[ r = \frac{360}{240} = 1.5 \]
-
From 360 to 540: \[ r = \frac{540}{360} = 1.5 \]
-
From 540 to 810: \[ r = \frac{810}{540} = 1.5 \]
The common ratio \( r \) is consistent at 1.5.
The formula for the \( n \)-th term of a geometric sequence is given by:
\[ f(n) = a_1 \cdot r^{n-1} \]
Substituting \( a_1 = 240 \) and \( r = 1.5 \):
\[ f(n) = 240 \cdot (1.5)^{n-1} \]
Now we can relate the options to this formula.
The only option that is close to this structure is:
- \( f(n) = 240 (12)^{n-1} \)
However, this does not match.
Since none of the provided options accurately reflect the form of \( f(n) = 240 \cdot (1.5)^{n-1} \), the correct response choice related clerical errors in transcription of choice is the best alternative.
Please ensure to verify the choices given as they don't seem to mathematically align with a typical geometric sequence equation. Should the intent be to represent the growth with an appropriate model, consider seeking a slightly modified version reflecting the correct base growth factor.
In conclusion, based on correct reasoning, you should have a function like:
\[ f(n) = 240 \cdot (1.5)^{n-1} \]
If options were not capturing this choice, a clarification on the formatting would be beneficial.