To determine the correct model for the population of the endangered species based on the given geometric sequence (240, 360, 540, 810, ...), we need to identify the common ratio.
- Calculating 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: \( \frac{810}{540} = 1.5 \)
The common ratio is \( r = 1.5 \).
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Modeling the function: The general form for a geometric sequence can be given by:
\[ f(n) = a \cdot r^{n-1} \] where \( a \) is the first term and \( r \) is the common ratio.
In this case:
- \( a = 240 \)
- \( r = 1.5 \)
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Substituting into the function: Therefore, the function that models this situation is:
\[ f(n) = 240 \cdot (1.5)^{n-1} \]
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Matching to the given options: In terms of the provided responses, we have:
- \( f(n) = 240(1.5)^{n-1} \) is equivalent to \( f(n) = 240(3/2)^{n-1} \).
However, since this specific option might not be directly listed, and we don't see \( 1.5 \) or \( 3/2 \) explicitly in your options, the closest equivalent is:
- f(n)=240(32)n−1
But since \( 32 \) seems incorrect, and the other options do not represent valid models based on the geometric growth inferred, it appears none of the provided options directly match the correct function.
To conclude, the correct model function based on the given terms is:
Final Answer: \[ f(n) = 240 \left( \frac{3}{2} \right)^{n-1} \] but since this isn't listed, ensure to recheck the options provided for accuracy.