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Answer :
To determine the recursive function used to generate the given arithmetic sequence, we need to identify the common difference and the first term of the sequence.
Here's the sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex]
### Step-by-Step Solution:
1. Identify the First Term:
- The first term of the sequence, denoted as [tex]\(f(1)\)[/tex], is the first number in the sequence.
- Here, [tex]\(f(1) = 14\)[/tex].
2. Find the Common Difference:
- An arithmetic sequence has a constant difference between consecutive terms. This is known as the common difference.
- To find the common difference, subtract the first term from the second term: [tex]\(24 - 14 = 10\)[/tex].
3. Write the Recursive Function:
- A recursive function for an arithmetic sequence can be written as [tex]\(f(n+1) = f(n) + d\)[/tex], where [tex]\(d\)[/tex] is the common difference and [tex]\(f(1)\)[/tex] is the first term.
- Based on our findings, the common difference is [tex]\(10\)[/tex], and the first term is [tex]\(14\)[/tex].
- Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Thus, the correct statement that describes the recursive function for the sequence is:
"The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
Here's the sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex]
### Step-by-Step Solution:
1. Identify the First Term:
- The first term of the sequence, denoted as [tex]\(f(1)\)[/tex], is the first number in the sequence.
- Here, [tex]\(f(1) = 14\)[/tex].
2. Find the Common Difference:
- An arithmetic sequence has a constant difference between consecutive terms. This is known as the common difference.
- To find the common difference, subtract the first term from the second term: [tex]\(24 - 14 = 10\)[/tex].
3. Write the Recursive Function:
- A recursive function for an arithmetic sequence can be written as [tex]\(f(n+1) = f(n) + d\)[/tex], where [tex]\(d\)[/tex] is the common difference and [tex]\(f(1)\)[/tex] is the first term.
- Based on our findings, the common difference is [tex]\(10\)[/tex], and the first term is [tex]\(14\)[/tex].
- Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
Thus, the correct statement that describes the recursive function for the sequence is:
"The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
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