Let's analyze the given sequence [tex]\(99.4, 0, -99.4, -198.8\)[/tex] to determine the appropriate recursive formula. We will denote the [tex]\(n\)[/tex]-th term of the sequence by [tex]\(f(n)\)[/tex].
1. Given Information:
[tex]\[
f(1) = 99.4, f(2) = 0, f(3) = -99.4, f(4) = -198.8
\][/tex]
2. Identifying the Pattern:
Let's calculate the differences between consecutive terms to identify the pattern:
[tex]\[
\begin{aligned}
f(2) - f(1) &= 0 - 99.4 = -99.4, \\
f(3) - f(2) &= -99.4 - 0 = -99.4, \\
f(4) - f(3) &= -198.8 - (-99.4) = -198.8 + 99.4 = -99.4.
\end{aligned}
\][/tex]
3. Formulating the Recursive Relation:
From the differences calculated above, we observe that the change between consecutive terms is consistent and equals [tex]\(-99.4\)[/tex]. This suggests a linear relationship between consecutive terms:
[tex]\[
f(n+1) = f(n) - 99.4 \quad \text{for} \quad n \geq 1
\][/tex]
4. Checking Options:
- Option 1: [tex]\( f(n+1) = f(n) + 99.4 \)[/tex]
[tex]\[
\text{This does not fit because the differences are negative, not positive.}
\][/tex]
- Option 2: [tex]\( f(n+1) = f(n) - 99.4 \)[/tex]
[tex]\[
\text{This matches the observed pattern.}
\][/tex]
- Option 3: [tex]\( f(n+1) = 99.4 f(n) \)[/tex]
[tex]\[
\text{This does not fit because the terms are not proportional to each other by a factor of 99.4.}
\][/tex]
- Option 4: [tex]\( f(n+1) = -99.4 f(n) \)[/tex]
[tex]\[
\text{This does not fit because the terms are not proportional to each other by a factor of -99.4.}
\][/tex]
Therefore, the recursive formula that correctly represents the sequence is:
[tex]\[
f(n+1) = f(n) - 99.4 \quad \text{for} \quad n \geq 1
\][/tex]
Hence, the correct choice is:
[tex]\[
f(n+1) = f(n) - 99.4, \, n \geq 1
\][/tex]
