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Answer :
To solve this problem, we need to identify the pattern of the arithmetic sequence and determine which statement correctly describes the recursive function for the sequence. Here's a step-by-step breakdown:
1. Identify the Sequence and Calculate Differences:
- The given sequence is: 4, 24, 34, 44, 54, ...
2. Calculate the Differences Between Consecutive Terms:
- Difference between the second term (24) and the first term (4): [tex]\( 24 - 4 = 20 \)[/tex]
- Difference between the third term (34) and the second term (24): [tex]\( 34 - 24 = 10 \)[/tex]
- Difference between the fourth term (44) and the third term (34): [tex]\( 44 - 34 = 10 \)[/tex]
3. Confirm the Common Difference:
- Notice that starting from the second difference, the differences are consistent at 10. This suggests the sequence from the second term onward has a consistent pattern with a common difference of 10.
4. Determine the Recursive Function:
- Since the consistent common difference from the second term onward is 10, and the sequence starting from the first term (4) follows this pattern, the recursive function that describes this sequence is:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
- The first term, which is [tex]\( f(1) \)[/tex], is 4.
5. Select the Correct Statement:
- Among the options provided, the statement that describes this situation is:
- "The common difference is 10, so the function is [tex]\( f(n+1)=f(n)+10 \)[/tex] where [tex]\( f(1)=4 \)[/tex]."
This describes the recursive relationship of the sequence accurately.
1. Identify the Sequence and Calculate Differences:
- The given sequence is: 4, 24, 34, 44, 54, ...
2. Calculate the Differences Between Consecutive Terms:
- Difference between the second term (24) and the first term (4): [tex]\( 24 - 4 = 20 \)[/tex]
- Difference between the third term (34) and the second term (24): [tex]\( 34 - 24 = 10 \)[/tex]
- Difference between the fourth term (44) and the third term (34): [tex]\( 44 - 34 = 10 \)[/tex]
3. Confirm the Common Difference:
- Notice that starting from the second difference, the differences are consistent at 10. This suggests the sequence from the second term onward has a consistent pattern with a common difference of 10.
4. Determine the Recursive Function:
- Since the consistent common difference from the second term onward is 10, and the sequence starting from the first term (4) follows this pattern, the recursive function that describes this sequence is:
[tex]\[ f(n+1) = f(n) + 10 \][/tex]
- The first term, which is [tex]\( f(1) \)[/tex], is 4.
5. Select the Correct Statement:
- Among the options provided, the statement that describes this situation is:
- "The common difference is 10, so the function is [tex]\( f(n+1)=f(n)+10 \)[/tex] where [tex]\( f(1)=4 \)[/tex]."
This describes the recursive relationship of the sequence accurately.
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