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Answer :
To find the recursive rule for the given arithmetic sequence [tex]\(12, 27, 42, 57, \ldots\)[/tex], let's go through the steps:
1. Identify the First Term:
The first term of the sequence is given as [tex]\(f(1) = 12\)[/tex].
2. Determine the Common Difference:
To find the common difference, we subtract the first term from the second term.
[tex]\[
\text{Second term} - \text{First term} = 27 - 12 = 15
\][/tex]
Therefore, the common difference [tex]\(d = 15\)[/tex].
3. Form the Recursive Rule:
In an arithmetic sequence, each term is obtained by adding the common difference to the previous term. The recursive formula expresses this relationship.
Since we know the first term [tex]\(f(1) = 12\)[/tex] and the common difference [tex]\(d = 15\)[/tex], the recursive rule for the sequence is:
[tex]\[
f(n) = f(n-1) + 15 \quad \text{for } n \geq 2
\][/tex]
Combining these, the recursive rule for the arithmetic sequence is:
- [tex]\(f(1) = 12\)[/tex]
- [tex]\(f(n) = f(n-1) + 15, \, n \geq 2\)[/tex]
Among the given options, the correct choice is:
[tex]\(f(1) = 12, \, f(n) = f(n-1) + 15, \, n \geq 2\)[/tex]
1. Identify the First Term:
The first term of the sequence is given as [tex]\(f(1) = 12\)[/tex].
2. Determine the Common Difference:
To find the common difference, we subtract the first term from the second term.
[tex]\[
\text{Second term} - \text{First term} = 27 - 12 = 15
\][/tex]
Therefore, the common difference [tex]\(d = 15\)[/tex].
3. Form the Recursive Rule:
In an arithmetic sequence, each term is obtained by adding the common difference to the previous term. The recursive formula expresses this relationship.
Since we know the first term [tex]\(f(1) = 12\)[/tex] and the common difference [tex]\(d = 15\)[/tex], the recursive rule for the sequence is:
[tex]\[
f(n) = f(n-1) + 15 \quad \text{for } n \geq 2
\][/tex]
Combining these, the recursive rule for the arithmetic sequence is:
- [tex]\(f(1) = 12\)[/tex]
- [tex]\(f(n) = f(n-1) + 15, \, n \geq 2\)[/tex]
Among the given options, the correct choice is:
[tex]\(f(1) = 12, \, f(n) = f(n-1) + 15, \, n \geq 2\)[/tex]
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