To determine which sequence could be partially defined by the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex] for [tex]\( n \geq 1 \)[/tex], we need to check the difference between successive terms in each sequence to see if it matches 2.5.
Let's analyze each sequence step-by-step:
### Sequence 1: [tex]\(2.5, 6.25, 15.625, 39.0625, \ldots\)[/tex]
- Difference between 6.25 and 2.5:
[tex]\( 6.25 - 2.5 = 3.75 \)[/tex]
- Difference between 15.625 and 6.25:
[tex]\( 15.625 - 6.25 = 9.375 \)[/tex]
- Difference between 39.0625 and 15.625:
[tex]\( 39.0625 - 15.625 = 23.4375 \)[/tex]
The differences are [tex]\(3.75, 9.375, 23.4375\)[/tex], which are not equal to 2.5. Therefore, the first sequence does not follow the given recursive formula.
### Sequence 2: [tex]\(2.5, 5, 10, 20\)[/tex]
- Difference between 5 and 2.5:
[tex]\( 5 - 2.5 = 2.5 \)[/tex]
- Difference between 10 and 5:
[tex]\( 10 - 5 = 5 \)[/tex]
- Difference between 20 and 10:
[tex]\( 20 - 10 = 10 \)[/tex]
The differences are [tex]\(2.5, 5, 10\)[/tex], which are not consistently 2.5. Thus, the second sequence does not follow the given recursive formula.
### Sequence 3: [tex]\(-10, -7.5, -5, -2.5, \ldots\)[/tex]
- Difference between [tex]\(-7.5\)[/tex] and [tex]\(-10\)[/tex]:
[tex]\( -7.5 - (-10) = -7.5 + 10 = 2.5 \)[/tex]
- Difference between [tex]\(-5\)[/tex] and [tex]\(-7.5\)[/tex]:
[tex]\( -5 - (-7.5) = -5 + 7.5 = 2.5 \)[/tex]
- Difference between [tex]\(-2.5\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\( -2.5 - (-5) = -2.5 + 5 = 2.5 \)[/tex]
The differences are [tex]\(2.5, 2.5, 2.5\)[/tex], which are consistently 2.5. Thus, the third sequence follows the given recursive formula.
### Sequence 4: [tex]\(-10, -25, 62.5, 156.25\)[/tex]
- Difference between [tex]\(-25\)[/tex] and [tex]\(-10\)[/tex]:
[tex]\( -25 - (-10) = -25 + 10 = -15 \)[/tex]
- Difference between 62.5 and [tex]\(-25\)[/tex]:
[tex]\( 62.5 - (-25) = 62.5 + 25 = 87.5 \)[/tex]
- Difference between 156.25 and 62.5:
[tex]\( 156.25 - 62.5 = 93.75 \)[/tex]
The differences are [tex]\(-15, 87.5, 93.75\)[/tex], which are not equal to 2.5. Therefore, the fourth sequence does not follow the given recursive formula.
### Conclusion
The third sequence, [tex]\(-10, -7.5, -5, -2.5, \ldots\)[/tex], is the only sequence that could be partially defined by the recursive formula [tex]\( f(n+1) = f(n) + 2.5 \)[/tex] for [tex]\( n \geq 1 \)[/tex] because the difference between successive terms is consistently 2.5.
