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)=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 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)=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 f(1)=−3; f(n)=f(an−1)+4

Answer 1

To determine the correct function notation for the recursively defined sequence given by \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), let’s analyze the options.

The recursive relationship indicates:
- The base case is \( a_1 = -3 \).
- Each subsequent term \( a_n \) is obtained by adding 4 to the previous term \( a_{n-1} \).

In function notation, we can express the sequence similarly. The relationship would be:
- \( f(1) = -3 \) which corresponds to the base case.
- The recursive formula can be written as \( f(n) = f(n-1) + 4 \) for \( n > 1 \), where \( f(n) \) defines the same relationship.

Now, looking at the options provided, we can identify the valid one based on this reasoning:

- **The correct response would be:**
\( f(1) = -3; \, f(n) = f(n-1) + 4 \)

This represents the base case and the recursive step accurately.