Given the recursively defined formula a1=−3

Question

, an=an−1+4
, determine which option is the correct function notation.(1 point)
Responses

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
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

Answer 1

The given recursively defined formula is:

• \( a_1 = -3 \)
• \( a_n = a_{n-1} + 4 \)

To express this in function notation, we start by letting \( f(n) \) represent the sequence defined by the recursive formula. Based on the definition:

1. The initial value is \( f(1) = a_1 = -3 \).
2. The recursive step can be written as \( f(n) = f(n-1) + 4 \).

So, looking at the options provided:

• The first choice describes the initial condition and the recursive step correctly: \[ a(1) = -3;, f(n) = f(n-1) + 4 \]

• The second choice has an incorrect recursive formula \( f(n) = a_{n-1} + 4 \).

• The third choice also has an incorrect recursive formula, \( f(n) = f(a_{n-1}) + 4 \).

• The fourth choice is identical to the first and is correct as well.

Both the first and fourth options are valid. If we are to choose only one in a typical scenario, we would typically select the first option.

Thus, the correct function notation is:

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