Answer :
To solve this problem, let's identify the type of sequence given: [tex]\(34, 40, 46, 52\)[/tex].
First, let's determine whether the sequence is arithmetic or geometric:
1. Arithmetic Sequence:
- An arithmetic sequence has a constant difference between consecutive terms. This difference is known as the common difference.
- To find the common difference, we subtract the first term from the second term: [tex]\(40 - 34 = 6\)[/tex].
- We can confirm if this difference is constant by checking the next differences:
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the common difference is always [tex]\(6\)[/tex], this sequence is arithmetic.
2. Geometric Sequence:
- A geometric sequence has a constant ratio between consecutive terms.
- The ratio can be found by dividing the second term by the first term: [tex]\(\frac{40}{34} \approx 1.176\)[/tex].
Since the ratio is not constant for the next pairs of terms (and it doesn't match the pattern observed), the sequence is not geometric.
Next, we need to write the recursive function for an arithmetic sequence:
- The first term [tex]\(f(1) = 34\)[/tex].
- The common difference [tex]\(d = 6\)[/tex].
- The recursive function for an arithmetic sequence is given by:
[tex]\[
f(n) = f(n-1) + d \quad \text{for} \quad n \geq 2
\][/tex]
So, the recursive function will be:
[tex]\[
f(n) = f(n-1) + 6 \quad \text{with} \quad f(1) = 34
\][/tex]
Thus, the correct type of sequence and recursive function is:
arithmetic sequence; [tex]\( f(1)=34 ; f(n)=f(n-1)+6 \)[/tex], for [tex]\( n \geq 2\)[/tex]
First, let's determine whether the sequence is arithmetic or geometric:
1. Arithmetic Sequence:
- An arithmetic sequence has a constant difference between consecutive terms. This difference is known as the common difference.
- To find the common difference, we subtract the first term from the second term: [tex]\(40 - 34 = 6\)[/tex].
- We can confirm if this difference is constant by checking the next differences:
- [tex]\(46 - 40 = 6\)[/tex]
- [tex]\(52 - 46 = 6\)[/tex]
Since the common difference is always [tex]\(6\)[/tex], this sequence is arithmetic.
2. Geometric Sequence:
- A geometric sequence has a constant ratio between consecutive terms.
- The ratio can be found by dividing the second term by the first term: [tex]\(\frac{40}{34} \approx 1.176\)[/tex].
Since the ratio is not constant for the next pairs of terms (and it doesn't match the pattern observed), the sequence is not geometric.
Next, we need to write the recursive function for an arithmetic sequence:
- The first term [tex]\(f(1) = 34\)[/tex].
- The common difference [tex]\(d = 6\)[/tex].
- The recursive function for an arithmetic sequence is given by:
[tex]\[
f(n) = f(n-1) + d \quad \text{for} \quad n \geq 2
\][/tex]
So, the recursive function will be:
[tex]\[
f(n) = f(n-1) + 6 \quad \text{with} \quad f(1) = 34
\][/tex]
Thus, the correct type of sequence and recursive function is:
arithmetic sequence; [tex]\( f(1)=34 ; f(n)=f(n-1)+6 \)[/tex], for [tex]\( n \geq 2\)[/tex]