Sure! Let's take this step-by-step to derive the recursive formula for the sequence given:
1. Identify the terms of the sequence:
The sequence provided is [tex]\(6, 1, -4, -9, -14, \ldots\)[/tex].
2. Determine the initial value:
We are given that [tex]\( f(1) = 6 \)[/tex].
3. Find the common difference:
- The difference between the second term (1) and the first term (6) is [tex]\( 1 - 6 = -5 \)[/tex].
- The difference between the third term (-4) and the second term (1) is [tex]\( -4 - 1 = -5 \)[/tex].
- The difference between the fourth term (-9) and the third term (-4) is [tex]\( -9 - (-4) = -9 + 4 = -5 \)[/tex].
- The difference between the fifth term (-14) and the fourth term (-9) is [tex]\( -14 - (-9) = -14 + 9 = -5 \)[/tex].
We observe that the common difference between consecutive terms is [tex]\(-5\)[/tex].
4. Determine the recurrence relation:
- From the common difference, we can generate the next term by subtracting 5 from the current term.
- This translates to the formula: [tex]\( f(n+1) = f(n) - 5 \)[/tex].
5. Verify the other choices:
- [tex]\( f(n+1) = f(n) + 5 \)[/tex]:
This would imply the sequence is increasing by 5 each time, which doesn't match our given sequence.
- [tex]\( f(n) = f(n+1) - 5 \)[/tex]:
Rearranging this gives [tex]\( f(n+1) = f(n) + 5 \)[/tex], again implying the sequence is increasing, which is incorrect.
- [tex]\( f(n+1) = -5 f(n) \)[/tex]:
This would imply each term is -5 times the previous term, which does not hold true for our sequence, e.g., [tex]\(6 \ast (-5) \neq 1\)[/tex].
Thus, the correct recursive formula that generates the given sequence is:
[tex]\[ f(n+1) = f(n) - 5 \][/tex]
This correctly describes the pattern observed in the sequence.
