To determine which sequence could be generated using the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex], we need to investigate each sequence separately.
### Sequence Analysis:
1. Sequence: [tex]\(1, -5, 25, -125, \ldots\)[/tex]
- The differences between the terms are: [tex]\( -5 - 1 = -6 \)[/tex], [tex]\( 25 - (-5) = 30 \)[/tex], and [tex]\( -125 - 25 = -150 \)[/tex].
- These differences do not follow the pattern described by the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex].
2. Sequence: [tex]\(2, 10, 50, 250, \ldots\)[/tex]
- The differences between the terms are: [tex]\( 10 - 2 = 8 \)[/tex], [tex]\( 50 - 10 = 40 \)[/tex], and [tex]\( 250 - 50 = 200 \)[/tex].
- These differences do not follow the pattern described by the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex].
3. Sequence: [tex]\(3, -2, -7, -12, \ldots\)[/tex]
- The differences between the terms are: [tex]\( -2 - 3 = -5 \)[/tex], [tex]\( -7 - (-2) = -5 \)[/tex], and [tex]\( -12 - (-7) = -5 \)[/tex].
- These differences exactly match the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex].
4. Sequence: [tex]\(4, 9, 14, 19, \ldots\)[/tex]
- The differences between the terms are: [tex]\( 9 - 4 = 5 \)[/tex], [tex]\( 14 - 9 = 5 \)[/tex], and [tex]\( 19 - 14 = 5 \)[/tex].
- These differences do not follow the pattern described by the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex].
Based on the analysis of the differences between consecutive terms in each sequence, the sequence that fits the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex] is:
[tex]\[ 3, -2, -7, -12, \ldots \][/tex]
Therefore, the sequence that could be generated using the recursive formula [tex]\( f(n+1) = f(n) - 5 \)[/tex] is:
[tex]\[ 3, -2, -7, -12, \ldots \][/tex]
