Let's analyze the given sequence: [tex]$3, -6, 12, -24, 48, \ldots$[/tex]
Identify [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3 \][/tex]
Now, let's observe the pattern in the sequence by calculating the subsequent terms.
1. To calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = -6 \][/tex]
2. To calculate [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = 12 \][/tex]
3. To calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = -24 \][/tex]
4. To calculate [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = 48 \][/tex]
We can look at these calculations to find the relationship between successive terms:
[tex]\[ f(2) = -6 = -3 \times 3 = -3 \times f(1) \][/tex]
[tex]\[ f(3) = 12 = -3 \times -6 = -3 \times f(2) \][/tex]
[tex]\[ f(4) = -24 = -3 \times 12 = -3 \times f(3) \][/tex]
[tex]\[ f(5) = 48 = -3 \times -24 = -3 \times f(4) \][/tex]
From these calculations, we observe a pattern:
[tex]\[ f(n+1) = -3 \times f(n) \][/tex]
Therefore, the recursive formula that can be used to generate the sequence is:
[tex]\[ f(n+1) = -3 f(n) \][/tex]
So the correct choice is:
[tex]\[ \boxed{f(n+1)=-3 f(n)} \][/tex]
