- Identify the sequence as arithmetic with first term \(f(1) = 5\) and common difference \(d = -6\).
- Apply the recursive definition for an arithmetic sequence \(f(n+1) = f(n) + d\).
- Substitute \(d = -6\) to obtain \(f(n+1) = f(n) - 6\).
- Verify the formula matches the given sequence terms.
- Final answer: \(\boxed{f(n+1) = f(n) - 6}\).
Explanation
- Analyzing the Sequence
We are given the sequence \(5, -1, -7, -13, -19, \ldots\) and we are told that \(f(1) = 5\). To understand the pattern, we examine the differences between consecutive terms:
\(f(2) - f(1) = -1 - 5 = -6\)
\(f(3) - f(2) = -7 - (-1) = -6\)
\(f(4) - f(3) = -13 - (-7) = -6\)
Since the difference between consecutive terms is consistently \(-6\), we recognize this as an arithmetic sequence with a common difference \(d = -6\).
- Determining the Recursive Formula
An arithmetic sequence with a constant difference \(d\) can be defined recursively as \(f(n+1) = f(n) + d\).
In our case, the common difference is \(d = -6\). Substituting this into the recursive formula, we get:
\(f(n+1) = f(n) - 6\).
We can verify this formula by testing it with our initial term:
For \(n=1\): \(f(2) = f(1) - 6 = 5 - 6 = -1\)
For \(n=2\): \(f(3) = f(2) - 6 = -1 - 6 = -7\)
These results match the given terms in the sequence.
- Selecting the Correct Option
Comparing our derived formula \(f(n+1) = f(n) - 6\) with the given options:
- \(f(n+1) = f(n) + 6\)
- \(f(n) = f(n+1) - 6 \implies f(n+1) = f(n) + 6\)
- \(f(n+1) = f(n) - 6\)
- \(f(n+1) = -6f(n)\)
The correct formula is indeed \(f(n+1) = f(n) - 6\), which corresponds to the third option. Thus, the final answer is \(\boxed{f(n+1) = f(n) - 6}\).
Examples
Recursive formulas are incredibly useful in finance and computer science. For example, if you are saving money in an account where you withdraw a fixed amount of \(\$6\) each month for a recurring subscription service, you can model your account balance using a recursive formula. If your starting balance is \(\$100\), the sequence of your balance \(B(n)\) after \(n\) months would be \(B(n+1) = B(n) - 6\). This helps you predict exactly when your balance will drop to zero or reach a specific threshold.