To find the first five terms in the given arithmetic sequence using the recursion formula, we proceed step-by-step according to the given rules.
The formula provided is:
[tex]\[ f(1) = 3 \][/tex]
[tex]\[ f(n) = f(n-1) - 2 \][/tex]
### Step-by-Step Calculation:
1. First term:
Given [tex]\( f(1) = 3 \)[/tex].
2. Second term:
To find [tex]\( f(2) \)[/tex], we use the recursive formula:
[tex]\[ f(2) = f(1) - 2 \][/tex]
Substituting [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = 3 - 2 \][/tex]
[tex]\[ f(2) = 1 \][/tex]
3. Third term:
Now, we find [tex]\( f(3) \)[/tex] using the recursive formula:
[tex]\[ f(3) = f(2) - 2 \][/tex]
Substituting [tex]\( f(2) \)[/tex]:
[tex]\[ f(3) = 1 - 2 \][/tex]
[tex]\[ f(3) = -1 \][/tex]
4. Fourth term:
Next, we determine [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = f(3) - 2 \][/tex]
Substituting [tex]\( f(3) \)[/tex]:
[tex]\[ f(4) = -1 - 2 \][/tex]
[tex]\[ f(4) = -3 \][/tex]
5. Fifth term:
Finally, for [tex]\( f(5) \)[/tex]:
[tex]\[ f(5) = f(4) - 2 \][/tex]
Substituting [tex]\( f(4) \)[/tex]:
[tex]\[ f(5) = -3 - 2 \][/tex]
[tex]\[ f(5) = -5 \][/tex]
### Conclusion:
The first five terms of the sequence are:
[tex]\[ 3, 1, -1, -3, -5 \][/tex]
So, the correct sequence is given by:
[tex]\[ \boxed{3, 1, -1, -3, -5} \][/tex]
