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Answer :
To determine the type of sequence and the recursive function for the sequence [tex]\(34, 40, 46, 52\)[/tex], let's follow these steps:
1. Identify the Type of Sequence:
- An arithmetic sequence has a constant difference between consecutive terms.
- A geometric sequence has a constant ratio between consecutive terms.
2. Find the Differences Between Terms:
- Calculate the difference between each pair of consecutive terms:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the differences are constant (all equal to 6), this sequence is an arithmetic sequence.
3. Determine the Recursive Function:
- For an arithmetic sequence, the recursive formula is:
[tex]\[
f(n) = f(n-1) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
- In this case, the common difference [tex]\(d\)[/tex] is 6, and the first term [tex]\(f(1)\)[/tex] is 34.
Therefore, the recursive function for this sequence is:
[tex]\[
f(1) = 34 ; \quad f(n) = f(n-1) + 6, \quad \text{for } n \geq 2
\][/tex]
Based on this analysis, the sequence is an arithmetic sequence, and the correct recursive function is:
[tex]\[ f(1)=34 ; \, f(n)=f(n-1)+6, \, \text{for } n \geq 2 \][/tex]
1. Identify the Type of Sequence:
- An arithmetic sequence has a constant difference between consecutive terms.
- A geometric sequence has a constant ratio between consecutive terms.
2. Find the Differences Between Terms:
- Calculate the difference between each pair of consecutive terms:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the differences are constant (all equal to 6), this sequence is an arithmetic sequence.
3. Determine the Recursive Function:
- For an arithmetic sequence, the recursive formula is:
[tex]\[
f(n) = f(n-1) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
- In this case, the common difference [tex]\(d\)[/tex] is 6, and the first term [tex]\(f(1)\)[/tex] is 34.
Therefore, the recursive function for this sequence is:
[tex]\[
f(1) = 34 ; \quad f(n) = f(n-1) + 6, \quad \text{for } n \geq 2
\][/tex]
Based on this analysis, the sequence is an arithmetic sequence, and the correct recursive function is:
[tex]\[ f(1)=34 ; \, f(n)=f(n-1)+6, \, \text{for } n \geq 2 \][/tex]
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