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Answer :
To solve this question, we need to understand the pattern of the arithmetic sequence given: 14, 24, 34, 44, 54, …
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the "common difference."
Let's determine the common difference for this sequence:
1. Start by subtracting the first term from the second term:
[tex]\(24 - 14 = 10\)[/tex].
2. Verify that the common difference is consistent by checking between other consecutive terms:
[tex]\(34 - 24 = 10\)[/tex],
[tex]\(44 - 34 = 10\)[/tex],
[tex]\(54 - 44 = 10\)[/tex].
In each case, the difference is 10, confirming that the common difference is indeed 10.
Now, let's establish the recursive function to generate this arithmetic sequence. A recursive function for an arithmetic sequence can be written as:
[tex]\[ f(n+1) = f(n) + \text{common difference} \][/tex]
Given that the first term [tex]\( f(1) \)[/tex] is 14, the recursive function specifically for this sequence would be:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
[tex]\[ f(1) = 14 \][/tex]
Therefore, the statement that accurately 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]."
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the "common difference."
Let's determine the common difference for this sequence:
1. Start by subtracting the first term from the second term:
[tex]\(24 - 14 = 10\)[/tex].
2. Verify that the common difference is consistent by checking between other consecutive terms:
[tex]\(34 - 24 = 10\)[/tex],
[tex]\(44 - 34 = 10\)[/tex],
[tex]\(54 - 44 = 10\)[/tex].
In each case, the difference is 10, confirming that the common difference is indeed 10.
Now, let's establish the recursive function to generate this arithmetic sequence. A recursive function for an arithmetic sequence can be written as:
[tex]\[ f(n+1) = f(n) + \text{common difference} \][/tex]
Given that the first term [tex]\( f(1) \)[/tex] is 14, the recursive function specifically for this sequence would be:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
[tex]\[ f(1) = 14 \][/tex]
Therefore, the statement that accurately 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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