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Answer :
Certainly! Let's analyze the arithmetic sequence provided:
The sequence is: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex].
1. Identify the First Term:
The first term of the sequence is [tex]\(14\)[/tex].
2. Determine the Common Difference:
In an arithmetic sequence, the common difference is the amount each term increases (or decreases) from the previous term.
- Subtract the first term from the second term:
[tex]\(24 - 14 = 10\)[/tex].
So, the common difference is [tex]\(10\)[/tex].
3. Write the Recursive Function:
A recursive function for an arithmetic sequence is generally written in the form [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.
For this sequence:
- The common difference [tex]\(d\)[/tex] is [tex]\(10\)[/tex].
- The first term [tex]\(f(1)\)[/tex] is [tex]\(14\)[/tex].
Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Conclusion:
The sequence has a common difference of [tex]\(10\)[/tex], and the recursive function used to generate the sequence is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex].
The sequence is: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex].
1. Identify the First Term:
The first term of the sequence is [tex]\(14\)[/tex].
2. Determine the Common Difference:
In an arithmetic sequence, the common difference is the amount each term increases (or decreases) from the previous term.
- Subtract the first term from the second term:
[tex]\(24 - 14 = 10\)[/tex].
So, the common difference is [tex]\(10\)[/tex].
3. Write the Recursive Function:
A recursive function for an arithmetic sequence is generally written in the form [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.
For this sequence:
- The common difference [tex]\(d\)[/tex] is [tex]\(10\)[/tex].
- The first term [tex]\(f(1)\)[/tex] is [tex]\(14\)[/tex].
Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Conclusion:
The sequence has a common difference of [tex]\(10\)[/tex], and the recursive function used to generate the sequence is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex].
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