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Answer :
To determine the type of sequence and the correct recursive function for the given sequence [tex]\(34, 40, 46, 52\)[/tex], let's break it down step by step:
### Step 1: Determine the Type of Sequence
1. Identify the Pattern:
- Look at the differences between consecutive terms to see if the sequence is arithmetic.
- Calculate the difference:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
2. Check the Differences:
- The differences between consecutive terms are consistent and equal to 6.
- This means the sequence is an arithmetic sequence because the difference is constant.
### Step 2: Write the Recursive Function
1. Identify the First Term:
- The first term of the sequence is [tex]\(f(1) = 34\)[/tex].
2. Formulate the Recursive Function:
- For an arithmetic sequence, the recursive function is based on the common difference. If each term is [tex]\(6\)[/tex] more than the previous term, then:
- [tex]\(f(n) = f(n-1) + 6\)[/tex] for [tex]\(n \geq 2\)[/tex].
### Conclusion
Based on the analysis, the given sequence [tex]\(34, 40, 46, 52\)[/tex] is an arithmetic sequence with a recursive function defined as follows:
- Type of Sequence: Arithmetic sequence
- Recursive Function:
- [tex]\(f(1) = 34\)[/tex]
- [tex]\(f(n) = f(n-1) + 6\)[/tex], for [tex]\(n \geq 2\)[/tex]
Thus, the correct answer is that the sequence is an arithmetic sequence with the recursive function: [tex]\(f(1)=34 ; f(n)=f(n-1)+6\)[/tex] for [tex]\(n \geq 2\)[/tex].
### Step 1: Determine the Type of Sequence
1. Identify the Pattern:
- Look at the differences between consecutive terms to see if the sequence is arithmetic.
- Calculate the difference:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
2. Check the Differences:
- The differences between consecutive terms are consistent and equal to 6.
- This means the sequence is an arithmetic sequence because the difference is constant.
### Step 2: Write the Recursive Function
1. Identify the First Term:
- The first term of the sequence is [tex]\(f(1) = 34\)[/tex].
2. Formulate the Recursive Function:
- For an arithmetic sequence, the recursive function is based on the common difference. If each term is [tex]\(6\)[/tex] more than the previous term, then:
- [tex]\(f(n) = f(n-1) + 6\)[/tex] for [tex]\(n \geq 2\)[/tex].
### Conclusion
Based on the analysis, the given sequence [tex]\(34, 40, 46, 52\)[/tex] is an arithmetic sequence with a recursive function defined as follows:
- Type of Sequence: Arithmetic sequence
- Recursive Function:
- [tex]\(f(1) = 34\)[/tex]
- [tex]\(f(n) = f(n-1) + 6\)[/tex], for [tex]\(n \geq 2\)[/tex]
Thus, the correct answer is that the sequence is an arithmetic sequence with the recursive function: [tex]\(f(1)=34 ; f(n)=f(n-1)+6\)[/tex] for [tex]\(n \geq 2\)[/tex].
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