To determine which recursive formula can be used to generate the sequence [tex]\(5, -1, -7, -13, -19, \ldots\)[/tex], where [tex]\(f(1) = 5\)[/tex] and [tex]\(n \geq 1\)[/tex], let’s analyze the behavior and pattern of the sequence.
Observing the given sequence:
- [tex]\(f(1) = 5\)[/tex]
- [tex]\(f(2) = -1\)[/tex]
- [tex]\(f(3) = -7\)[/tex]
- [tex]\(f(4) = -13\)[/tex]
- [tex]\(f(5) = -19\)[/tex]
First, identify the difference between consecutive terms of the sequence:
- [tex]\(f(2) - f(1) = -1 - 5 = -6\)[/tex]
- [tex]\(f(3) - f(2) = -7 - (-1) = -7 + 1 = -6\)[/tex]
- [tex]\(f(4) - f(3) = -13 - (-7) = -13 + 7 = -6\)[/tex]
- [tex]\(f(5) - f(4) = -19 - (-13) = -19 + 13 = -6\)[/tex]
From this, it's clear that the difference between consecutive terms is consistently [tex]\(-6\)[/tex].
Hence, the sequence decreases by 6 for each subsequent term. We need to express this pattern using a recursive formula. This can be directly translated as:
[tex]\[ f(n+1) = f(n) - 6 \][/tex]
This relation shows that each new term [tex]\(f(n+1)\)[/tex] is obtained by subtracting 6 from the previous term [tex]\(f(n)\)[/tex].
Now, let’s consider the given choices:
- [tex]\( f(n+1) = f(n) + 6 \)[/tex]
- [tex]\( f(n) = f(n+1) - 6 \)[/tex]
- [tex]\( f(n+1) = f(n) - 6 \)[/tex]
- [tex]\( f(n+1) = -6 f(n) \)[/tex]
The correct formulation that matches our analysis of the sequence is:
[tex]\[ f(n+1) = f(n) - 6 \][/tex]
Therefore, the recursive formula used to generate the given sequence is:
[tex]\[ \boxed{f(n+1) = f(n) - 6} \][/tex]
