Given the recursively defined formula a1=−3

Question

, 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

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

Answer 1

To determine the correct function notation based on the recursively defined formula for the sequence given, let's analyze the definitions provided:

1. We start with \( a_1 = -3 \).
2. The recursion defines each subsequent term as \( a_n = a_{n-1} + 4 \).

Given the choices, we are looking for a function notation that corresponds correctly to these definitions.

Let's analyze the options:

1. Option 1: \[ f(1) = -3 ; \quad f(n) = a_{n-1} + 4 \]

   This incorrectly uses \( a_{n-1} \), which does not align with the recursive definition.

2. Option 2: \[ f(1) = -3 ; \quad f(n) = f(a_{n-1}) + 4 \]

   This also incorrectly assumes \( f(n) \) is based on the output of another function \( f \) with \( a_{n-1} \).

3. Option 3: \[ f(1) = -3 ; \quad f(n) = f(n-1) + 4 \]

   This correctly defines \( f(n) \) based on the previous term \( f(n-1) \), mirroring the recursion.

4. Option 4: \[ a(1) = -3 ; \quad f(n) = f(n-1) + 4 \]

   This incorrectly uses \( a(1) \) and does not specify the sequence correctly since \( a(n) \) and \( f(n) \) are treated separately.

Among these options, Option 3 is the correct one because it maintains the recursive pattern of the original sequence perfectly:

Correct Answer: \[ f(1) = -3 ; \quad f(n) = f(n-1) + 4 \]