Ask a New Question
Menu

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

1 answer

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.