Question
The population of an endangered species was estimated to be 240 in 2018. The population has been tracked each year and has been modeled with the geometric sequence 240, 360, 540, 810, . . . . Create a function that models this situation.(1 point)
Responses
f(n)=240+(n−1)12
f left parenthesis n right parenthesis equals 240 plus left parenthesis n minus 1 right parenthesis Start Fraction 1 over 2 End Fraction
f(n)=240(120)n−1
f left parenthesis n right parenthesis equals 240 left parenthesis 120 right parenthesis superscript n minus 1 baseline
f(n)=240(12)n−1
f left parenthesis n right parenthesis equals 240 left parenthesis Start Fraction 1 over 2 End Fraction right parenthesis superscript n minus 1 baseline
f(n)=240(32)n−1
Responses
f(n)=240+(n−1)12
f left parenthesis n right parenthesis equals 240 plus left parenthesis n minus 1 right parenthesis Start Fraction 1 over 2 End Fraction
f(n)=240(120)n−1
f left parenthesis n right parenthesis equals 240 left parenthesis 120 right parenthesis superscript n minus 1 baseline
f(n)=240(12)n−1
f left parenthesis n right parenthesis equals 240 left parenthesis Start Fraction 1 over 2 End Fraction right parenthesis superscript n minus 1 baseline
f(n)=240(32)n−1
Answers
GPT-4o mini
To model the population of the endangered species with a function, we first need to analyze the pattern in the population values given as a geometric sequence: 240, 360, 540, 810, ...
To determine the common ratio, we can observe how each term relates to the previous one:
1. From 240 to 360, the increase factor is:
\[
\frac{360}{240} = 1.5
\]
2. From 360 to 540, the increase factor is:
\[
\frac{540}{360} = 1.5
\]
3. From 540 to 810, the increase factor is:
\[
\frac{810}{540} = 1.5
\]
Thus, the common ratio \( r \) is 1.5.
A geometric sequence can be modeled by the formula:
\[
f(n) = a \cdot r^{n-1}
\]
where \( a \) is the first term and \( r \) is the common ratio.
In this case, \( a = 240 \) and \( r = 1.5 \). Therefore, the function that models this situation is:
\[
f(n) = 240 \cdot (1.5)^{n-1}
\]
Now, let's look at the given options. The correct function among the responses should be:
\[
f(n) = 240(1.5)^{n-1}
\]
This matches the geometric sequence with a starting term of 240 and a common ratio of 1.5.
However, none of the provided response options are formatted correctly for \( r = 1.5 \).
You may have made a typo in stating the options, or there could be an oversight regarding the specified values. Be sure that numerical values and common ratios reflect the correct geometric model.
To determine the common ratio, we can observe how each term relates to the previous one:
1. From 240 to 360, the increase factor is:
\[
\frac{360}{240} = 1.5
\]
2. From 360 to 540, the increase factor is:
\[
\frac{540}{360} = 1.5
\]
3. From 540 to 810, the increase factor is:
\[
\frac{810}{540} = 1.5
\]
Thus, the common ratio \( r \) is 1.5.
A geometric sequence can be modeled by the formula:
\[
f(n) = a \cdot r^{n-1}
\]
where \( a \) is the first term and \( r \) is the common ratio.
In this case, \( a = 240 \) and \( r = 1.5 \). Therefore, the function that models this situation is:
\[
f(n) = 240 \cdot (1.5)^{n-1}
\]
Now, let's look at the given options. The correct function among the responses should be:
\[
f(n) = 240(1.5)^{n-1}
\]
This matches the geometric sequence with a starting term of 240 and a common ratio of 1.5.
However, none of the provided response options are formatted correctly for \( r = 1.5 \).
You may have made a typo in stating the options, or there could be an oversight regarding the specified values. Be sure that numerical values and common ratios reflect the correct geometric model.