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Answer :
To solve the problem of identifying the recursive function for the given arithmetic sequence, let's break down the steps:
1. Identify the terms of the sequence:
The sequence is: 14, 24, 34, 44, 54, ...
2. Calculate the common difference:
To find the common difference, subtract the first term from the second term:
[tex]\[
\text{Common difference} = 24 - 14 = 10
\][/tex]
To verify, subtract the second term from the third term:
[tex]\[
34 - 24 = 10
\][/tex]
The common difference is consistent at 10 between consecutive terms.
3. Determine the first term of the sequence:
The first term in the sequence [tex]\( f(1) \)[/tex] is 14.
4. Formulate the recursive function:
Based on the common difference and the first term, the recursive function for the sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
This function successfully describes how to generate each term in the sequence. Therefore, the statement "The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex]." is the correct description.
1. Identify the terms of the sequence:
The sequence is: 14, 24, 34, 44, 54, ...
2. Calculate the common difference:
To find the common difference, subtract the first term from the second term:
[tex]\[
\text{Common difference} = 24 - 14 = 10
\][/tex]
To verify, subtract the second term from the third term:
[tex]\[
34 - 24 = 10
\][/tex]
The common difference is consistent at 10 between consecutive terms.
3. Determine the first term of the sequence:
The first term in the sequence [tex]\( f(1) \)[/tex] is 14.
4. Formulate the recursive function:
Based on the common difference and the first term, the recursive function for the sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
This function successfully describes how to generate each term in the sequence. Therefore, the statement "The common difference is 10, so the function is [tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex]." is the correct description.
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