To determine which sequence could be defined by the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex] for [tex]\( n \geq 1 \)[/tex], we need to check each sequence and see if it follows the rule that each term is obtained by adding 2.5 to the previous term.
Let's analyze each sequence in turn:
1. Sequence: [tex]\(2.5, 6.25, 15.625, 39.0625, \ldots\)[/tex]
- [tex]\( f(2) = 6.25 \)[/tex]
- [tex]\( f(3) = 15.625 \)[/tex]
- [tex]\( f(4) = 39.0625 \)[/tex]
Let's check if [tex]\( f(n+1) = f(n) + 2.5 \)[/tex]:
- [tex]\( 2.5 + 2.5 = 5 \neq 6.25 \)[/tex]
- This sequence does not follow the recursive formula.
2. Sequence: [tex]\(2.5, 5, 10, 20, \ldots\)[/tex]
- [tex]\( f(2) = 5 \)[/tex]
- [tex]\( f(3) = 10 \)[/tex]
- [tex]\( f(4) = 20 \)[/tex]
Let's check if [tex]\( f(n+1) = f(n) + 2.5 \)[/tex]:
- [tex]\( 2.5 + 2.5 = 5 \)[/tex]
- [tex]\( 5 + 2.5 = 7.5 \neq 10 \)[/tex]
- This sequence does not follow the recursive formula.
3. Sequence: [tex]\(-10, -7.5, -5, -2.5, \ldots\)[/tex]
- [tex]\( f(2) = -7.5 \)[/tex]
- [tex]\( f(3) = -5 \)[/tex]
- [tex]\( f(4) = -2.5 \)[/tex]
Let's check if [tex]\( f(n+1) = f(n) + 2.5 \)[/tex]:
- [tex]\(-10 + 2.5 = -7.5 \)[/tex]
- [tex]\(-7.5 + 2.5 = -5 \)[/tex]
- [tex]\(-5 + 2.5 = -2.5 \)[/tex]
- This sequence follows the recursive formula.
4. Sequence: [tex]\(-10, -25, 62.5, 156.25, \ldots\)[/tex]
- [tex]\( f(2) = -25 \)[/tex]
- [tex]\( f(3) = 62.5 \)[/tex]
- [tex]\( f(4) = 156.25 \)[/tex]
Let's check if [tex]\( f(n+1) = f(n) + 2.5 \)[/tex]:
- [tex]\(-10 + 2.5 = -7.5 \neq -25 \)[/tex]
- This sequence does not follow the recursive formula.
Based on the assessment, only the third sequence [tex]\((-10, -7.5, -5, -2.5, \ldots)\)[/tex] follows the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex].
Therefore, the sequence that could be partially defined by the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
