- The recursive formula \(f(n+1) = f(n) - 5\) describes an arithmetic sequence with a common difference of \(-5\).
- We test each given option by calculating the difference between consecutive terms.
- The sequence \(3, -2, -7, -12, \dots\) consistently maintains a difference of \(-5\) between consecutive terms.
- The correct sequence is \(\boxed{3, -2, -7, -12, \dots}\).
Explanation
- Analysis of the Problem
The recursive formula \(f(n+1) = f(n) - 5\) defines an arithmetic sequence where each term is obtained by subtracting 5 from the previous term. This means the common difference between consecutive terms must be exactly -5. We need to identify which of the given sequences satisfies this property. To test this, we calculate the difference between consecutive terms (\(f(n+1) - f(n)\)) for each provided sequence and check if it equals -5.
- Checking the Sequences
Let's examine each sequence provided:
- For the sequence \(1, -5, 25, -125, \dots\):
The difference between the second and first term is \(-5 - 1 = -6\), which is not -5.
- For the sequence \(2, 10, 50, 250, \dots\):
The difference between the second and first term is \(10 - 2 = 8\), which is not -5.
- For the sequence \(3, -2, -7, -12, \dots\):
The difference between the second and first term is \(-2 - 3 = -5\).
The difference between the third and second term is \(-7 - (-2) = -5\).
The difference between the fourth and third term is \(-12 - (-7) = -5\).
Since the difference is constantly -5, this sequence follows the recursive rule.
- For the sequence \(4, 9, 14, 19, \dots\):
The difference between the second and first term is \(9 - 4 = 5\), which is not -5.
- Conclusion
After evaluating each sequence, it is clear that only the sequence \(3, -2, -7, -12, \dots\) maintains a constant difference of -5 between consecutive terms. Thus, this is the only sequence generated by the formula \(f(n+1) = f(n) - 5\). The correct sequence is \(\boxed{3, -2, -7, -12, \dots}\).
Examples
Recursive sequences are fundamental for modeling systems that change incrementally. For example, consider a savings account where you withdraw a fixed amount of \(\$5\) every month from an initial balance. If you start with \(\$3\), your balance will follow the sequence \(3, -2, -7, \dots\), showing how your balance changes over time. Understanding recursive formulas helps in predicting future values in such financial or logistical processes.