Let's analyze the sequence [tex]\(99.4, 0, -99.4, -198.8\)[/tex] to find the correct recursive formula.
1. Check the first formula: [tex]\( f(n+1) = f(n) + 99.4 \)[/tex]
- [tex]\( f(1) = 99.4 \)[/tex]
- [tex]\( f(2) = f(1) + 99.4 = 99.4 + 99.4 = 198.8 \)[/tex]
The second term should be [tex]\(0\)[/tex], but we get [tex]\(198.8\)[/tex]. So, this formula is incorrect.
2. Check the second formula: [tex]\( f(n+1) = f(n) - 99.4 \)[/tex]
- [tex]\( f(1) = 99.4 \)[/tex]
- [tex]\( f(2) = f(1) - 99.4 = 99.4 - 99.4 = 0 \)[/tex]
- [tex]\( f(3) = f(2) - 99.4 = 0 - 99.4 = -99.4 \)[/tex]
- [tex]\( f(4) = f(3) - 99.4 = -99.4 - 99.4 = -198.8 \)[/tex]
The sequence matches: [tex]\(99.4, 0, -99.4, -198.8\)[/tex]. Therefore, this is a potential correct formula.
3. Check the third formula: [tex]\( f(n+1) = 99.4 \cdot f(n) \)[/tex]
- [tex]\( f(1) = 99.4 \)[/tex]
- [tex]\( f(2) = 99.4 \cdot f(1) = 99.4 \cdot 99.4 = 9880.36 \)[/tex]
The second term should be [tex]\(0\)[/tex], but we get [tex]\(9880.36\)[/tex]. So, this formula is incorrect.
4. Check the fourth formula: [tex]\( f(n+1) = -99.4 \cdot f(n) \)[/tex]
- [tex]\( f(1) = 99.4 \)[/tex]
- [tex]\( f(2) = -99.4 \cdot f(1) = -99.4 \cdot 99.4 = -9880.36 \)[/tex]
The second term should be [tex]\(0\)[/tex], but we get [tex]\(-9880.36\)[/tex]. So, this formula is incorrect.
Upon inspection, the correct recursive formula for the sequence [tex]\(99.4, 0, -99.4, -198.8\)[/tex], where [tex]\( f(1) = 99.4 \)[/tex] is indeed:
[tex]\[ f(n+1) = f(n) - 99.4, \ n \geq 1 \][/tex]
