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Answer :
To determine the correct type of sequence and the recursive function for the sequence [tex]\(34, 40, 46, 52\)[/tex], let's analyze each option step by step.
### Step 1: Identify the Pattern
First, we need to identify any pattern in the sequence by looking at the differences between consecutive numbers:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the difference between consecutive terms is constant, the sequence has a common difference of 6. This indicates that the sequence is an arithmetic sequence.
### Step 2: Confirm Type of Sequence
An arithmetic sequence has a constant difference between consecutive terms, which we just confirmed. Therefore, the sequence is arithmetic.
### Step 3: Determine the Recursive Function
For an arithmetic sequence, the recursive function is generally of the form:
[tex]\[ f(n) = f(n-1) + d \][/tex]
where [tex]\(d\)[/tex] is the common difference.
In our sequence, the first term is 34, and the common difference [tex]\(d\)[/tex] is 6. Therefore, the recursive function can be expressed as:
- Starting term: [tex]\(f(1) = 34\)[/tex]
- Recursive formula: [tex]\(f(n) = f(n-1) + 6\)[/tex]
### Final Answer
From the given options, the sequence is an arithmetic sequence, and the correct recursive function is:
- Arithmetic sequence; [tex]\(f(1) = 34 ; f(n) = f(n-1) + 6\)[/tex], for [tex]\(n \geq 2\)[/tex].
This matches the second option provided.
### Step 1: Identify the Pattern
First, we need to identify any pattern in the sequence by looking at the differences between consecutive numbers:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the difference between consecutive terms is constant, the sequence has a common difference of 6. This indicates that the sequence is an arithmetic sequence.
### Step 2: Confirm Type of Sequence
An arithmetic sequence has a constant difference between consecutive terms, which we just confirmed. Therefore, the sequence is arithmetic.
### Step 3: Determine the Recursive Function
For an arithmetic sequence, the recursive function is generally of the form:
[tex]\[ f(n) = f(n-1) + d \][/tex]
where [tex]\(d\)[/tex] is the common difference.
In our sequence, the first term is 34, and the common difference [tex]\(d\)[/tex] is 6. Therefore, the recursive function can be expressed as:
- Starting term: [tex]\(f(1) = 34\)[/tex]
- Recursive formula: [tex]\(f(n) = f(n-1) + 6\)[/tex]
### Final Answer
From the given options, the sequence is an arithmetic sequence, and the correct recursive function is:
- Arithmetic sequence; [tex]\(f(1) = 34 ; f(n) = f(n-1) + 6\)[/tex], for [tex]\(n \geq 2\)[/tex].
This matches the second option provided.
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