To determine the correct recursive formula to generate the given sequence, we need to understand the pattern of the sequence. The sequence provided is:
[tex]\[ 6, 1, -4, -9, -14, \ldots \][/tex]
### Step-by-Step Analysis:
1. Identify the first term:
- The first term of the sequence is [tex]\( f(1) = 6 \)[/tex].
2. Calculate the common difference:
- To find the common difference, subtract consecutive terms:
[tex]\[
1 - 6 = -5 \\
-4 - 1 = -5 \\
-9 - (-4) = -5 \\
-14 - (-9) = -5
\][/tex]
- The common difference is [tex]\(-5\)[/tex].
3. Form the recursive formula:
- A recursive formula expresses each term based on the previous term.
- Given [tex]\( f(n) \)[/tex] is the current term, the next term [tex]\( f(n + 1) \)[/tex] can be expressed as:
[tex]\[
f(n + 1) = f(n) - 5
\][/tex]
- This formula correctly captures the pattern where each term is obtained by subtracting 5 from the previous term.
4. Verify other options:
- Check the remaining options to ensure we have the correct one:
[tex]\[
f(n + 1) = f(n) + 5 \quad \text{(Incorrect: this would increase terms by 5)} \\
f(n) = f(n + 1) - 5 \quad \text{(Incorrect: equivalent to } f(n + 1) = f(n) + 5 \text{)} \\
f(n + 1) = -5 f(n) \quad \text{(Incorrect: this would multiply terms by -5)}
\][/tex]
### Conclusion:
The correct recursive formula to generate the sequence where [tex]\( f(1) = 6 \)[/tex] and [tex]\( n \geq 1 \)[/tex] is:
[tex]\[ f(n + 1) = f(n) - 5 \][/tex]
This formula accurately describes the relationship between consecutive terms in the given sequence.
