To determine the appropriate recursive formula for the sequence [tex]\(5, -1, -7, -13, -19, \ldots \)[/tex], we need to observe how the terms change from one to the next.
First, let's note the initial values:
- The first term [tex]\( f(1) \)[/tex] is 5.
Next, let's find the difference between each consecutive term:
- The difference between the second term and the first term: [tex]\(-1 - 5 = -6\)[/tex].
- The difference between the third term and the second term: [tex]\(-7 - (-1) = -7 + 1 = -6\)[/tex].
- The difference between the fourth term and the third term: [tex]\(-13 - (-7) = -13 + 7 = -6\)[/tex].
- The difference between the fifth term and the fourth term: [tex]\(-19 - (-13) = -19 + 13 = -6\)[/tex].
We see that the sequence consistently decreases by 6 at each step. This means that to get the next term, we subtract 6 from the current term.
Therefore, the recursive formula that fits this pattern is:
[tex]\[ f(n+1) = f(n) - 6 \][/tex]
This means the correct formula to generate the given sequence is:
[tex]\[ \boxed{f(n+1) = f(n) - 6} \][/tex]
