Answer :
To determine the recursive rule for the sequence [tex]\(54, 60, 66, \ldots\)[/tex], let's break down the process step-by-step:
1. Identify the Starting Term:
- The sequence begins with the first term, which is [tex]\(54\)[/tex]. So, [tex]\(f(1) = 54\)[/tex].
2. Calculate the Common Difference:
- Find the difference between consecutive terms in the sequence to see if there is a consistent pattern.
- Calculate the difference between the second term and the first term: [tex]\(60 - 54 = 6\)[/tex].
- Similarly, confirm with the difference between the third term and the second term: [tex]\(66 - 60 = 6\)[/tex].
3. Determine the Recursive Rule:
- Having established that the difference between each successive term is [tex]\(6\)[/tex], we can express the sequence with a recursive rule.
- The recursive rule will add this common difference to the preceding term to find the next term.
- This means the sequence rule is: [tex]\(f(n) = f(n-1) + 6\)[/tex].
4. Final Recursive Rule:
- Given the starting term and the common difference, the recursive rule for the sequence is:
[tex]\[
f(1) = 54, \quad f(n) = f(n-1) + 6
\][/tex]
This explains why the correct recursive rule for the sequence is [tex]\(f(1) = 54, f(n) = f(n-1) + 6\)[/tex].
1. Identify the Starting Term:
- The sequence begins with the first term, which is [tex]\(54\)[/tex]. So, [tex]\(f(1) = 54\)[/tex].
2. Calculate the Common Difference:
- Find the difference between consecutive terms in the sequence to see if there is a consistent pattern.
- Calculate the difference between the second term and the first term: [tex]\(60 - 54 = 6\)[/tex].
- Similarly, confirm with the difference between the third term and the second term: [tex]\(66 - 60 = 6\)[/tex].
3. Determine the Recursive Rule:
- Having established that the difference between each successive term is [tex]\(6\)[/tex], we can express the sequence with a recursive rule.
- The recursive rule will add this common difference to the preceding term to find the next term.
- This means the sequence rule is: [tex]\(f(n) = f(n-1) + 6\)[/tex].
4. Final Recursive Rule:
- Given the starting term and the common difference, the recursive rule for the sequence is:
[tex]\[
f(1) = 54, \quad f(n) = f(n-1) + 6
\][/tex]
This explains why the correct recursive rule for the sequence is [tex]\(f(1) = 54, f(n) = f(n-1) + 6\)[/tex].
