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Which recursive formula can be used to define this sequence for ?Choices:(A) (B) (C) (D)
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Q. Which recursive formula can be used to define this sequence for ?Choices:(A) (B) (C) (D)
- Determine Type: Determine if the sequence is arithmetic or geometric.We need to check if the difference between consecutive terms is constant (which would make it an arithmetic sequence) or if the ratio between consecutive terms is constant (which would make it a geometric sequence).
- Calculate Difference: Calculate the difference between consecutive terms.The difference between the first two terms is . Let's check if this difference is consistent with the next pair of terms: . It appears that the difference is constant.
- Confirm Arithmetic: Confirm that the sequence is arithmetic. Since the difference between consecutive terms is constant, we can confirm that the sequence is arithmetic with a common difference of .
- Identify Recursive Formula: Identify the recursive formula for the given arithmetic sequence.For an arithmetic sequence, the recursive formula is generally of the form , where is the common difference. In this case, .
- Match with Choices: Match the recursive formula with the given choices.The correct recursive formula based on our findings is . Let's match this with the given choices.(A) is incorrect because it lacks the subscript to denote the term number.(B) is incorrect because it suggests a geometric sequence with a ratio, not an arithmetic sequence with a difference.(C) is incorrect because it suggests subtracting , not adding.(D) is incorrect because it suggests adding the previous term to itself and then adding , which is not the pattern we observed.The correct choice is not explicitly listed, but it is clear that the intended correct choice is (A) with the correct notation: .
