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Answer :
To determine the type of sequence and the correct recursive function, let's analyze the given 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.
- Calculate the differences between consecutive terms:
- [tex]\( 40 - 34 = 6 \)[/tex]
- [tex]\( 46 - 40 = 6 \)[/tex]
- [tex]\( 52 - 46 = 6 \)[/tex]
- Since the differences are all the same (6), the sequence is arithmetic with a common difference of 6.
2. Check if it's a Geometric Sequence:
- In a geometric sequence, the ratio between consecutive terms is constant.
- Calculate the ratios:
- [tex]\( \frac{40}{34} \)[/tex] does not equal [tex]\( \frac{46}{40} \)[/tex], and so on.
- Since the ratios are not constant, it's not a geometric sequence.
Step 2: Determine the Recursive Formula for the Arithmetic Sequence
An arithmetic sequence can be defined recursively. For this sequence:
- The first term is [tex]\( f(1) = 34 \)[/tex].
- The recursive formula for an arithmetic sequence is:
[tex]\[
f(n) = f(n-1) + \text{common difference}, \text{ for } n \geq 2
\][/tex]
- Given the common difference here is 6, the formula becomes:
[tex]\[
f(n) = f(n-1) + 6, \text{ for } n \geq 2
\][/tex]
Conclusion:
The correct answer is:
- It is an arithmetic sequence.
- The recursive function is:
[tex]\[
f(1) = 34; \quad f(n) = f(n-1) + 6, \text{ for } n \geq 2
\][/tex]
This matches the option "arithmetic sequence; [tex]\( f(1) = 34; f(n) = f(n-1) + 6 \)[/tex], for [tex]\( n \geq 2 \)[/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.
- Calculate the differences between consecutive terms:
- [tex]\( 40 - 34 = 6 \)[/tex]
- [tex]\( 46 - 40 = 6 \)[/tex]
- [tex]\( 52 - 46 = 6 \)[/tex]
- Since the differences are all the same (6), the sequence is arithmetic with a common difference of 6.
2. Check if it's a Geometric Sequence:
- In a geometric sequence, the ratio between consecutive terms is constant.
- Calculate the ratios:
- [tex]\( \frac{40}{34} \)[/tex] does not equal [tex]\( \frac{46}{40} \)[/tex], and so on.
- Since the ratios are not constant, it's not a geometric sequence.
Step 2: Determine the Recursive Formula for the Arithmetic Sequence
An arithmetic sequence can be defined recursively. For this sequence:
- The first term is [tex]\( f(1) = 34 \)[/tex].
- The recursive formula for an arithmetic sequence is:
[tex]\[
f(n) = f(n-1) + \text{common difference}, \text{ for } n \geq 2
\][/tex]
- Given the common difference here is 6, the formula becomes:
[tex]\[
f(n) = f(n-1) + 6, \text{ for } n \geq 2
\][/tex]
Conclusion:
The correct answer is:
- It is an arithmetic sequence.
- The recursive function is:
[tex]\[
f(1) = 34; \quad f(n) = f(n-1) + 6, \text{ for } n \geq 2
\][/tex]
This matches the option "arithmetic sequence; [tex]\( f(1) = 34; f(n) = f(n-1) + 6 \)[/tex], for [tex]\( n \geq 2 \)[/tex]".
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