To determine the recursive rule for the given sequence [tex]\(3, -2, -7, -12, \ldots\)[/tex], we need to identify the pattern between the terms. Here is a step-by-step approach to find the recursive formula:
1. Identify the common difference:
The given sequence is [tex]\(3, -2, -7, -12, \ldots\)[/tex]. Let's calculate the difference between consecutive terms:
- The difference between the second term [tex]\(-2\)[/tex] and the first term [tex]\(3\)[/tex]:
[tex]\[
-2 - 3 = -5
\][/tex]
- The difference between the third term [tex]\(-7\)[/tex] and the second term [tex]\(-2\)[/tex]:
[tex]\[
-7 - (-2) = -7 + 2 = -5
\][/tex]
- The difference between the fourth term [tex]\(-12\)[/tex] and the third term [tex]\(-7\)[/tex]:
[tex]\[
-12 - (-7) = -12 + 7 = -5
\][/tex]
We observe that the common difference, [tex]\(d\)[/tex], is consistently [tex]\(-5\)[/tex].
2. Formulate the recursive rule:
In the recursive rule for an arithmetic sequence, each term is obtained by adding the common difference to the previous term.
Since our common difference is [tex]\(-5\)[/tex], the recursive formula will subtract [tex]\(5\)[/tex] from the previous term to get the next term.
3. Write the recursive formula:
If [tex]\(f(n)\)[/tex] represents the [tex]\(n\)[/tex]-th term of the sequence, the recursive rule can be expressed as:
[tex]\[
f(n) = f(n-1) - 5
\][/tex]
Additionally, the starting term of the sequence is given:
[tex]\[
f(1) = 3
\][/tex]
Thus, the recursive rule for the sequence is:
[tex]\[
f(n) = f(n-1) - 5, \quad f(1) = 3
\][/tex]
Among the given options, the correct answer is:
[tex]\[
f(n) = f(n-1) - 5, \quad f(1) = 3
\][/tex]
