Answer :
Let's examine each of the options carefully to determine which functions accurately describe the number of miles recorded on the odometer after [tex]\( n \)[/tex] months, given that Natalie’s car reads 800 miles after the first month and she expects to drive an additional 900 miles each subsequent month.
1. [tex]\(\begin{array}{l} f(1)=900 \\ f(n)=f(n-1)+800, \text { for } n \geq 2 \end{array}\)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 900 \)[/tex]. But the odometer initially reads 800 miles, so this function's value is incorrect for [tex]\( f(1) \)[/tex].
- Therefore, this option is incorrect.
2. [tex]\(\begin{array}{l} f(1)=800 \\ f(n)=f(n-1)+900, \text { for } n \geq 2 \end{array}\)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 800 \)[/tex], which is correct.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = f(1) + 900 = 800 + 900 = 1700 \)[/tex]. This seems consistent with the problem statement.
- This option correctly represents the mileage as [tex]\( f(n) = 800 + 900(n-1) \)[/tex].
- Hence, this option is correct.
3. [tex]\( f(n) = 900n - 100 \)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 900 \times 1 - 100 = 800 \)[/tex], which matches the starting mileage.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = 900 \times 2 - 100 = 1700 \)[/tex], which is also consistent with the expected mileage.
- Therefore, this option is correct.
4. [tex]\( f(n) = 800n + 100 \)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 800 \times 1 + 100 = 900 \)[/tex], which does not match the initial reading of 800 miles.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = 800 \times 2 + 100 = 1700 \times 2 + 100 = 1700 \)[/tex]. Here the calculation for non-initial months is inconsistent.
- Hence, this option is incorrect.
5. [tex]\(\begin{array}{l} f(1) = 800 \\ f(n) = f(n-1) + 100, \text{ for } n \geq 2 \end{array}\)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 800 \)[/tex], which is correct.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = f(1) + 100 = 800 + 100 = 900 \)[/tex]. This is incorrect as per the given problem.
- Hence, this option is incorrect.
6. [tex]\( f(n) = 900n + 800 \)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 900 \times 1 + 800 = 1700 \)[/tex]. This does not match the initial reading.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = 900 \times 2 + 800 = 1700 + 900 = 2600 \)[/tex]. This does not match the problem statement either.
- Though the initial summary showed this option to be considered correct, let's reconsider the applied problem.
Given the consistent check on the matching problem states, the correctly described mileages are only fitting with chosen functions optionally 1, 2, and 6 must be derived with varying consistent mileage calculation.
Therefore, the correct answers are:
- Option 2: [tex]\( \begin{array}{l} f(1)=800 \\ f(n)=f(n-1)+900, \text { for } n \geq 2 \end{array} \)[/tex]
- Option 3: [tex]\( f(n)=900 n-100 \)[/tex]
1. [tex]\(\begin{array}{l} f(1)=900 \\ f(n)=f(n-1)+800, \text { for } n \geq 2 \end{array}\)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 900 \)[/tex]. But the odometer initially reads 800 miles, so this function's value is incorrect for [tex]\( f(1) \)[/tex].
- Therefore, this option is incorrect.
2. [tex]\(\begin{array}{l} f(1)=800 \\ f(n)=f(n-1)+900, \text { for } n \geq 2 \end{array}\)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 800 \)[/tex], which is correct.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = f(1) + 900 = 800 + 900 = 1700 \)[/tex]. This seems consistent with the problem statement.
- This option correctly represents the mileage as [tex]\( f(n) = 800 + 900(n-1) \)[/tex].
- Hence, this option is correct.
3. [tex]\( f(n) = 900n - 100 \)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 900 \times 1 - 100 = 800 \)[/tex], which matches the starting mileage.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = 900 \times 2 - 100 = 1700 \)[/tex], which is also consistent with the expected mileage.
- Therefore, this option is correct.
4. [tex]\( f(n) = 800n + 100 \)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 800 \times 1 + 100 = 900 \)[/tex], which does not match the initial reading of 800 miles.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = 800 \times 2 + 100 = 1700 \times 2 + 100 = 1700 \)[/tex]. Here the calculation for non-initial months is inconsistent.
- Hence, this option is incorrect.
5. [tex]\(\begin{array}{l} f(1) = 800 \\ f(n) = f(n-1) + 100, \text{ for } n \geq 2 \end{array}\)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 800 \)[/tex], which is correct.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = f(1) + 100 = 800 + 100 = 900 \)[/tex]. This is incorrect as per the given problem.
- Hence, this option is incorrect.
6. [tex]\( f(n) = 900n + 800 \)[/tex]
- For [tex]\( n = 1 \)[/tex], [tex]\( f(1) = 900 \times 1 + 800 = 1700 \)[/tex]. This does not match the initial reading.
- For [tex]\( n = 2 \)[/tex], [tex]\( f(2) = 900 \times 2 + 800 = 1700 + 900 = 2600 \)[/tex]. This does not match the problem statement either.
- Though the initial summary showed this option to be considered correct, let's reconsider the applied problem.
Given the consistent check on the matching problem states, the correctly described mileages are only fitting with chosen functions optionally 1, 2, and 6 must be derived with varying consistent mileage calculation.
Therefore, the correct answers are:
- Option 2: [tex]\( \begin{array}{l} f(1)=800 \\ f(n)=f(n-1)+900, \text { for } n \geq 2 \end{array} \)[/tex]
- Option 3: [tex]\( f(n)=900 n-100 \)[/tex]
