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Answer :
To solve the problem of determining the recursive function for the given arithmetic sequence, let's break it down step-by-step:
1. Identify the Sequence:
The sequence given is:
[tex]\[14, 24, 34, 44, 54, \ldots\][/tex]
2. Determine the Common Difference:
Since this is an arithmetic sequence, the difference between consecutive terms is constant.
To find this difference, subtract the first term from the second term:
[tex]\[24 - 14 = 10\][/tex]
3. Write the Recursive Function:
An arithmetic sequence can be described by the recursive function:
[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 of the sequence.
4. Apply the Identified Values:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
Replacing these values into the recursive function gives us:
[tex]\[f(n+1) = f(n) + 10\][/tex]
with [tex]\(f(1) = 14\)[/tex].
Therefore, the correct statement that describes the recursive function used to generate 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]."
1. Identify the Sequence:
The sequence given is:
[tex]\[14, 24, 34, 44, 54, \ldots\][/tex]
2. Determine the Common Difference:
Since this is an arithmetic sequence, the difference between consecutive terms is constant.
To find this difference, subtract the first term from the second term:
[tex]\[24 - 14 = 10\][/tex]
3. Write the Recursive Function:
An arithmetic sequence can be described by the recursive function:
[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 of the sequence.
4. Apply the Identified Values:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.
Replacing these values into the recursive function gives us:
[tex]\[f(n+1) = f(n) + 10\][/tex]
with [tex]\(f(1) = 14\)[/tex].
Therefore, the correct statement that describes the recursive function used to generate 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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