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Answer :
To solve this problem, we need to identify the pattern in the given arithmetic sequence and then describe the recursive function that represents it. The sequence given is:
14, 24, 34, 44, 54, ...
Here's a step-by-step approach to understanding the sequence and finding the correct recursive function:
1. Identify the Common Difference:
- An arithmetic sequence is defined by a constant difference between consecutive terms.
- Let's find the difference between the first two terms: 24 - 14 = 10.
- Check the difference between other consecutive terms to ensure consistency:
- 34 - 24 = 10
- 44 - 34 = 10
- 54 - 44 = 10
- The common difference is consistently 10.
2. Define the First Term:
- The first term of the sequence, [tex]\( f(1) \)[/tex], is 14.
3. Describe the Recursive Function:
- A recursive function that generates this sequence takes the form:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- With the common difference identified as 10, the recursive function becomes:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Since the first term is 14, we have:
[tex]\[
f(1) = 14
\][/tex]
4. Select the Correct Statement:
- Based on the common difference of 10 and the first term being 14, the right statement 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 accurately describes the recursive function used to generate the arithmetic sequence given in the problem.
14, 24, 34, 44, 54, ...
Here's a step-by-step approach to understanding the sequence and finding the correct recursive function:
1. Identify the Common Difference:
- An arithmetic sequence is defined by a constant difference between consecutive terms.
- Let's find the difference between the first two terms: 24 - 14 = 10.
- Check the difference between other consecutive terms to ensure consistency:
- 34 - 24 = 10
- 44 - 34 = 10
- 54 - 44 = 10
- The common difference is consistently 10.
2. Define the First Term:
- The first term of the sequence, [tex]\( f(1) \)[/tex], is 14.
3. Describe the Recursive Function:
- A recursive function that generates this sequence takes the form:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- With the common difference identified as 10, the recursive function becomes:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]
- Since the first term is 14, we have:
[tex]\[
f(1) = 14
\][/tex]
4. Select the Correct Statement:
- Based on the common difference of 10 and the first term being 14, the right statement 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 accurately describes the recursive function used to generate the arithmetic sequence given in the problem.
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