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Answer :
To solve this problem, we need to identify the pattern in the arithmetic sequence and determine the recursive function that generates it.
Let's break it down step by step:
1. Identify the Sequence:
The sequence provided is: 14, 24, 34, 44, 54,...
2. Find the Common Difference:
- To find the common difference, subtract the first term from the second term:
[tex]\( 24 - 14 = 10 \)[/tex]
- It is a common arithmetic sequence step to find that all consecutive terms have the same difference. Let's verify:
[tex]\( 34 - 24 = 10 \)[/tex]
[tex]\( 44 - 34 = 10 \)[/tex]
[tex]\( 54 - 44 = 10 \)[/tex]
- This confirms the common difference is 10.
3. Define the Recursive Function:
- In a recursive function for an arithmetic sequence, you use the formula:
[tex]\( f(n+1) = f(n) + d \)[/tex]
Where [tex]\( d \)[/tex] is the common difference.
- Here, [tex]\( d = 10 \)[/tex] and the starting term [tex]\( f(1) = 14 \)[/tex].
4. Write the Recursive Function Statement:
- The recursive function that describes this sequence is:
[tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
From the options given in the initial question, the statement that accurately describes the recursive function for 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]."
Let's break it down step by step:
1. Identify the Sequence:
The sequence provided is: 14, 24, 34, 44, 54,...
2. Find the Common Difference:
- To find the common difference, subtract the first term from the second term:
[tex]\( 24 - 14 = 10 \)[/tex]
- It is a common arithmetic sequence step to find that all consecutive terms have the same difference. Let's verify:
[tex]\( 34 - 24 = 10 \)[/tex]
[tex]\( 44 - 34 = 10 \)[/tex]
[tex]\( 54 - 44 = 10 \)[/tex]
- This confirms the common difference is 10.
3. Define the Recursive Function:
- In a recursive function for an arithmetic sequence, you use the formula:
[tex]\( f(n+1) = f(n) + d \)[/tex]
Where [tex]\( d \)[/tex] is the common difference.
- Here, [tex]\( d = 10 \)[/tex] and the starting term [tex]\( f(1) = 14 \)[/tex].
4. Write the Recursive Function Statement:
- The recursive function that describes this sequence is:
[tex]\( f(n+1) = f(n) + 10 \)[/tex] where [tex]\( f(1) = 14 \)[/tex].
From the options given in the initial question, the statement that accurately describes the recursive function for 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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