Answer :
To solve this problem, we need to determine the common difference in the arithmetic sequence and use it to find the recursive function that generates the sequence. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is the same. This difference is called the common difference.
Let's examine the given sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex]
1. Identify the Pattern:
- First, we calculate the difference between the first and second terms: [tex]\(24 - 14 = 10\)[/tex].
- Next, calculate the difference between the second and third terms: [tex]\(34 - 24 = 10\)[/tex].
- Similarly, the difference between the third and fourth terms is: [tex]\(44 - 34 = 10\)[/tex], and between the fourth and fifth terms is: [tex]\(54 - 44 = 10\)[/tex].
2. Verify the Common Difference:
- From the calculations above, it's clear that the common difference is consistent and equal to 10.
3. Formulate the Recursive Function:
- In a recursive function for an arithmetic sequence, each term can be generated by adding the common difference to the previous term.
- Therefore, the recursive function describing this sequence is:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Additionally, the first term of the sequence is 14, so we can write:
[tex]\[
f(1) = 14
\][/tex]
4. Conclusion:
- The correct statement that describes the recursive function used to generate the sequence is:
[tex]\[
\text{The common difference is 10, so the function is } f(n+1) = f(n) + 10 \text{ where } f(1) = 14.
\][/tex]
This step-by-step approach shows how to identify the common difference and how it leads to the formulation of the recursive function for the sequence.
Let's examine the given sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex]
1. Identify the Pattern:
- First, we calculate the difference between the first and second terms: [tex]\(24 - 14 = 10\)[/tex].
- Next, calculate the difference between the second and third terms: [tex]\(34 - 24 = 10\)[/tex].
- Similarly, the difference between the third and fourth terms is: [tex]\(44 - 34 = 10\)[/tex], and between the fourth and fifth terms is: [tex]\(54 - 44 = 10\)[/tex].
2. Verify the Common Difference:
- From the calculations above, it's clear that the common difference is consistent and equal to 10.
3. Formulate the Recursive Function:
- In a recursive function for an arithmetic sequence, each term can be generated by adding the common difference to the previous term.
- Therefore, the recursive function describing this sequence is:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Additionally, the first term of the sequence is 14, so we can write:
[tex]\[
f(1) = 14
\][/tex]
4. Conclusion:
- The correct statement that describes the recursive function used to generate the sequence is:
[tex]\[
\text{The common difference is 10, so the function is } f(n+1) = f(n) + 10 \text{ where } f(1) = 14.
\][/tex]
This step-by-step approach shows how to identify the common difference and how it leads to the formulation of the recursive function for the sequence.
