Answer :
To determine the type of sequence and the recursive function for the sequence provided, let's examine the sequence: 34, 40, 46, 52.
### Step 1: Identify the Type of Sequence
1. Check if it's an Arithmetic Sequence:
- In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference.
- Let's calculate the differences between consecutive terms:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the difference is the same (6) for each pair of consecutive terms, the sequence is an arithmetic sequence.
2. Check if it's a Geometric Sequence (Optional Step):
- In a geometric sequence, each term is found by multiplying the previous one by a constant ratio.
- To check, we would divide consecutive terms to find a consistent ratio. However, given that we've already identified it as arithmetic, we don't need to proceed with this check.
### Step 2: Write the Recursive Function
Since we have an arithmetic sequence, we'll express it as a recursive function:
- The first term [tex]\(f(1)\)[/tex] is 34.
- The recursive rule for an arithmetic sequence can be written as:
[tex]\[
f(n) = f(n-1) + d, \quad \text{for } n \geq 2
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
In this specific sequence:
- The first term [tex]\(f(1)\)[/tex] is 34.
- The common difference [tex]\(d\)[/tex] is 6.
Thus, 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]
### Conclusion
The sequence [tex]\(34, 40, 46, 52\)[/tex] is an arithmetic sequence with the recursive function:
[tex]\[
f(1) = 34; \quad f(n) = f(n-1) + 6, \quad \text{for } n \geq 2
\][/tex]
### Step 1: Identify the Type of Sequence
1. Check if it's an Arithmetic Sequence:
- In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the common difference.
- Let's calculate the differences between consecutive terms:
- [tex]\(40 - 34 = 6\)[/tex]
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the difference is the same (6) for each pair of consecutive terms, the sequence is an arithmetic sequence.
2. Check if it's a Geometric Sequence (Optional Step):
- In a geometric sequence, each term is found by multiplying the previous one by a constant ratio.
- To check, we would divide consecutive terms to find a consistent ratio. However, given that we've already identified it as arithmetic, we don't need to proceed with this check.
### Step 2: Write the Recursive Function
Since we have an arithmetic sequence, we'll express it as a recursive function:
- The first term [tex]\(f(1)\)[/tex] is 34.
- The recursive rule for an arithmetic sequence can be written as:
[tex]\[
f(n) = f(n-1) + d, \quad \text{for } n \geq 2
\][/tex]
where [tex]\(d\)[/tex] is the common difference.
In this specific sequence:
- The first term [tex]\(f(1)\)[/tex] is 34.
- The common difference [tex]\(d\)[/tex] is 6.
Thus, 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]
### Conclusion
The sequence [tex]\(34, 40, 46, 52\)[/tex] is an arithmetic sequence with the recursive function:
[tex]\[
f(1) = 34; \quad f(n) = f(n-1) + 6, \quad \text{for } n \geq 2
\][/tex]