Given the recursively defined sequence where \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), we want to find the correct function notation among the options provided.
- Identifying the function:
- The initial condition is \( a_1 = -3 \).
- The recursive definition states that each term can be found by adding 4 to the previous term.
The options can be analyzed as follows:
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Option 1: \( f(1) = -3; f(n) = a_{n-1} + 4 \)
- Here, it defines \( f(1) = -3 \) correctly, but there's an inconsistency as \( f(n) \) uses \( a_{n-1} \) instead of \( f(n-1) \).
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Option 2: \( a(1) = -3; f(n) = f(n - 1) + 4 \)
- This option has a syntax issue; it uses \( a(1) \) instead of a consistent function notation for both cases.
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Option 3: \( f(1) = -3; f(n) = f(n- 1) + 4 \)
- This option correctly defines a function \( f \) to match the recursive definition. It has the correct base case and recursion.
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Option 4: \( f = -3; f(n) = f(a_{n - (a_{n-1})} + 4 \)
- This option doesn't correctly match the desired recursive definition, and also mixes up the notation.
Conclusion: The option that correctly represents the recursive sequence with appropriate function notation is:
Option 3: \( f(1) = -3; f(n) = f(n- 1) + 4 \)