Question

Given the recursively defined formula a1 = -3, an = an-1 + 4, determine which option is
the correct function notation. (1 point)
• f (1) = -3; f (n) = an-1 + 4
• a (1) = -3; f (n) = f(n - 1) + 4
• f(1) = -3; f(n) = f(n- 1) + 4
• f = -3; f (n) = f (an- (an-1) + 4

Answer 1

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.

1. **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:

1. **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 \( f(n-1) \).

2. **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.

3. **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.

4. **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**: \( f(1) = -3; f(n) = f(n- 1) + 4 \)