Answer :
To determine the recursive rule for the sequence [tex]\(54, 57, 60, 63, 66, \ldots\)[/tex], we need to first identify the pattern or rule that governs how the sequence progresses.
1. Identify the Starting Point:
The first term of the sequence is given as [tex]\(54\)[/tex].
2. Calculate the Differences Between Terms:
To find the pattern, observe the difference between consecutive terms:
- Subtract the first term from the second term: [tex]\(57 - 54 = 3\)[/tex].
- Subtract the second term from the third term: [tex]\(60 - 57 = 3\)[/tex].
- Subtract the third term from the fourth term: [tex]\(63 - 60 = 3\)[/tex].
- Subtract the fourth term from the fifth term: [tex]\(66 - 63 = 3\)[/tex].
Each difference is [tex]\(3\)[/tex], indicating that the sequence increases by [tex]\(3\)[/tex] each time.
3. Formulate the Recursive Rule:
With a common difference of [tex]\(3\)[/tex], the rule that describes how to get from one term to the next is:
- Start with [tex]\(f(1) = 54\)[/tex].
- For any term [tex]\(n\)[/tex], [tex]\(f(n) = f(n-1) + 3\)[/tex].
Thus, the recursive rule is:
[tex]\[
\text{(A) } f(1) = 54, \quad f(n) = f(n-1) + 3
\][/tex]
1. Identify the Starting Point:
The first term of the sequence is given as [tex]\(54\)[/tex].
2. Calculate the Differences Between Terms:
To find the pattern, observe the difference between consecutive terms:
- Subtract the first term from the second term: [tex]\(57 - 54 = 3\)[/tex].
- Subtract the second term from the third term: [tex]\(60 - 57 = 3\)[/tex].
- Subtract the third term from the fourth term: [tex]\(63 - 60 = 3\)[/tex].
- Subtract the fourth term from the fifth term: [tex]\(66 - 63 = 3\)[/tex].
Each difference is [tex]\(3\)[/tex], indicating that the sequence increases by [tex]\(3\)[/tex] each time.
3. Formulate the Recursive Rule:
With a common difference of [tex]\(3\)[/tex], the rule that describes how to get from one term to the next is:
- Start with [tex]\(f(1) = 54\)[/tex].
- For any term [tex]\(n\)[/tex], [tex]\(f(n) = f(n-1) + 3\)[/tex].
Thus, the recursive rule is:
[tex]\[
\text{(A) } f(1) = 54, \quad f(n) = f(n-1) + 3
\][/tex]
