Question

Given the recursively defined formula a1=−3, an=an−1+4, determine which option is the correct function notation.(1 point)
Responses

f(1)=−3; f(n)=f(an−1)+4
f left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals f left parenthesis a subscript n minus 1 baseline right parenthesis plus 4

f(1)=−3; f(n)=an−1+4
f left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals a subscript n minus 1 baseline plus 4

f(1)=−3; f(n)=f(n−1)+4
f left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals f left parenthesis n minus 1 right parenthesis plus 4

a(1)=−3; f(n)=f(n−1)+4
a left parenthesis 1 right parenthesis equals negative 3 ; f left parenthesis n right parenthesis equals f left parenthesis n minus 1 right parenthesis plus 4

Answer 1

To determine the correct function notation from the given options based on the recursively defined formula \( a_1 = -3 \) and \( a_n = a_{n−1} + 4 \), let's analyze the recursive definition.

1. The first term is defined as \( a_1 = -3 \).
2. For subsequent terms, \( a_n \) is defined recursively as \( a_n = a_{n-1} + 4 \).

Thus, the sequence starts at -3 and each subsequent term increases by 4.

Now, let's evaluate each option in the responses:

1. **Option 1**: \( f(1) = -3; f(n) = f(a_{n−1}) + 4 \)
- This is incorrect because the formula for \( f(n) \) does not align with the sequence definition. It incorrectly uses \( f(a_{n-1}) \) instead of \( f(n-1) \).

2. **Option 2**: \( f(1) = -3; f(n) = a_{n−1} + 4 \)
- This is also incorrect because it misuses the notation by not applying the function recursively (it should use \( f(n-1) \) rather than \( a_{n-1} \)).

3. **Option 3**: \( f(1) = -3; f(n) = f(n−1) + 4 \)
- This is the correct representation of the recursive formula since it matches the structure where each term is defined based on the previous term: \( f(n) = f(n-1) + 4 \). This captures the nature of recursion correctly.

4. **Option 4**: \( a(1) = -3; f(n) = f(n−1) + 4 \)
- This option is incorrect as it mixes the definitions of \( a \) and \( f \).

Based on this analysis, the correct answer is:

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