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)=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

f(1)=−3
; f(n)=an−1+4

Answer 1

To determine the correct function notation based on the recursively defined formula given, we can break it down.

1. The first value of the sequence is defined as \( a_1 = -3 \).
2. The recursive formula states that \( a_n = a_{n-1} + 4 \).

Let's look at the options provided:

1. **Option 1**:
\( f(1) = -3 \)
\( f(n) = f(a_{n-1}) + 4 \)
2. **Option 2**:
\( f(1) = -3 \)
\( f(n) = f(n-1) + 4 \)
3. **Option 3**:
\( a(1) = -3 \)
\( f(n) = f(n-1) + 4 \)
4. **Option 4**:
\( f(1) = -3 \)
\( f(n) = a_{n-1} + 4 \)

Now evaluating these:

- **Option 1** introduces \( f(a_{n-1}) \) which does not match \( a_n \)’s recursive relationship.
- **Option 2** correctly defines the recursive relationship using \( f(n) = f(n-1) + 4 \). This mirrors the behavior of the sequence where each term is the previous term plus 4.
- **Option 3** incorrectly uses \( a(1) \) instead of \( f(1) \) and states a relationship for \( f\) which is not equivalent to \( a_n\).
- **Option 4** states \( f(n) = a_{n-1} + 4 \), but does not use the recursive nature of \( f\).

Given these evaluations, **Option 2** is the correct function notation:

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