To determine the correct recursive formula for the given sequence [tex]\(3, -6, 12, -24, 48, \ldots\)[/tex], we start by examining how each term in the sequence relates to its predecessor.
Let's look at the sequence term by term:
1. 1st term: 3
2. 2nd term: -6
To get from the 1st term to the 2nd term, we multiply the 1st term by [tex]\(-2\)[/tex]:
[tex]\[ 3 \times (-2) = -6 \][/tex]
3. 3rd term: 12
To get from the 2nd term to the 3rd term, we multiply the 2nd term by [tex]\(-2\)[/tex]:
[tex]\[ -6 \times (-2) = 12 \][/tex]
4. 4th term: -24
To get from the 3rd term to the 4th term, we multiply the 3rd term by [tex]\(-2\)[/tex]:
[tex]\[ 12 \times (-2) = -24 \][/tex]
5. 5th term: 48
To get from the 4th term to the 5th term, we multiply the 4th term by [tex]\(-2\)[/tex]:
[tex]\[ -24 \times (-2) = 48 \][/tex]
The pattern is evident: each term is obtained by multiplying the previous term by [tex]\(-2\)[/tex]. Therefore, the recursive formula for the sequence is:
[tex]\[ f(n+1) = -2 f(n) \][/tex]
Given the options:
- [tex]\(f(n+1) = -3 f(n)\)[/tex]
- [tex]\(f(n+1) = 3 f(n)\)[/tex]
- [tex]\(f(n+1) = -2 f(n)\)[/tex]
- [tex]\(f(n+1) = 2 f(n)\)[/tex]
The correct recursive formula that generates the sequence [tex]\(3, -6, 12, -24, 48, \ldots\)[/tex] is:
[tex]\[ f(n+1) = -2 f(n) \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{f(n+1) = -2 f(n)} \][/tex]
