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Answer :
To determine which statement describes the recursive function used to generate the sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], we need to identify the common difference between consecutive terms and the first term of the sequence.
1. Identify the common difference:
To find the common difference, subtract the first term from the second term:
[tex]\[ 24 - 14 = 10 \][/tex]
Thus, the common difference is [tex]\(10\)[/tex].
2. Verify the common difference with other terms:
To ensure consistency, let's subtract other consecutive terms to confirm the common difference is indeed [tex]\(10\)[/tex]:
[tex]\[ 34 - 24 = 10 \][/tex]
[tex]\[ 44 - 34 = 10 \][/tex]
[tex]\[ 54 - 44 = 10 \][/tex]
Since the common difference remains [tex]\(10\)[/tex] for all pairs of consecutive terms, we can be confident that the common difference is [tex]\(10\)[/tex].
3. Identify the first term:
From the given sequence, the first term is clearly [tex]\(14\)[/tex].
4. Formulate the recursive function:
Using the common difference and the first term, we can write the recursive function as:
[tex]\[ f(n + 1) = f(n) + 10 \][/tex]
where [tex]\( f(1) = 14 \)[/tex].
Thus, the correct 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].
1. Identify the common difference:
To find the common difference, subtract the first term from the second term:
[tex]\[ 24 - 14 = 10 \][/tex]
Thus, the common difference is [tex]\(10\)[/tex].
2. Verify the common difference with other terms:
To ensure consistency, let's subtract other consecutive terms to confirm the common difference is indeed [tex]\(10\)[/tex]:
[tex]\[ 34 - 24 = 10 \][/tex]
[tex]\[ 44 - 34 = 10 \][/tex]
[tex]\[ 54 - 44 = 10 \][/tex]
Since the common difference remains [tex]\(10\)[/tex] for all pairs of consecutive terms, we can be confident that the common difference is [tex]\(10\)[/tex].
3. Identify the first term:
From the given sequence, the first term is clearly [tex]\(14\)[/tex].
4. Formulate the recursive function:
Using the common difference and the first term, we can write the recursive function as:
[tex]\[ f(n + 1) = f(n) + 10 \][/tex]
where [tex]\( f(1) = 14 \)[/tex].
Thus, the correct 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].
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