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Answer :
To determine the recursive function used to generate the given arithmetic sequence, let's analyze the sequence step by step.
The sequence is: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex].
1. Identify the common difference:
- To find the common difference in an arithmetic sequence, subtract a term from the next term.
- For this sequence, subtract the first term from the second term: [tex]\(24 - 14 = 10\)[/tex].
- This pattern continues across the sequence, so the common difference is [tex]\(10\)[/tex].
2. Determine the starting point:
- The first term of the sequence is [tex]\(14\)[/tex]. Hence, [tex]\(f(1) = 14\)[/tex].
3. Write the recursive formula:
- A recursive function for an arithmetic sequence is expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- For our sequence, this becomes:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
where the initial term [tex]\(f(1) = 14\)[/tex].
4. Conclusion:
- The correct statement describing the recursive function is: "The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
This step-by-step breakdown provides a clear understanding of how the recursive function is derived from the sequence.
The sequence is: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex].
1. Identify the common difference:
- To find the common difference in an arithmetic sequence, subtract a term from the next term.
- For this sequence, subtract the first term from the second term: [tex]\(24 - 14 = 10\)[/tex].
- This pattern continues across the sequence, so the common difference is [tex]\(10\)[/tex].
2. Determine the starting point:
- The first term of the sequence is [tex]\(14\)[/tex]. Hence, [tex]\(f(1) = 14\)[/tex].
3. Write the recursive formula:
- A recursive function for an arithmetic sequence is expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- For our sequence, this becomes:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
where the initial term [tex]\(f(1) = 14\)[/tex].
4. Conclusion:
- The correct statement describing the recursive function is: "The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
This step-by-step breakdown provides a clear understanding of how the recursive function is derived from the sequence.
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