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Answer :
To solve this problem, we need to analyze the arithmetic sequence given: 14, 24, 34, 44, 54, ...
1. Identify the Common Difference:
- In an arithmetic sequence, the difference between consecutive terms is constant. This is known as the common difference.
- To find the common difference, subtract the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
- So, the common difference is 10.
2. Create the Recursive Function:
- A recursive function for an arithmetic sequence can be expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- Here, we already determined the common difference is 10.
3. Determine the Initial Term:
- The first term of the sequence is given as 14, which is represented as [tex]\( f(1) = 14 \)[/tex].
4. Select the Correct Statement:
- From the given statements, the one that correctly describes the recursive function for this 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]."
This statement accurately captures the recursive function necessary to generate the sequence 14, 24, 34, 44, 54, etc.
1. Identify the Common Difference:
- In an arithmetic sequence, the difference between consecutive terms is constant. This is known as the common difference.
- To find the common difference, subtract the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
- So, the common difference is 10.
2. Create the Recursive Function:
- A recursive function for an arithmetic sequence can be expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- Here, we already determined the common difference is 10.
3. Determine the Initial Term:
- The first term of the sequence is given as 14, which is represented as [tex]\( f(1) = 14 \)[/tex].
4. Select the Correct Statement:
- From the given statements, the one that correctly describes the recursive function for this 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]."
This statement accurately captures the recursive function necessary to generate the sequence 14, 24, 34, 44, 54, etc.
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