Which recursive formula can be used to define this sequence for n 1? 2, 6, 10, 14, 18, 22, ... Choices: (A)a = a + a + 4 (B)a = 7/3a (C)a = a - 4 (D)a = a + 4

Question

Which recursive formula can be used to define this sequence for n > 1?   2, 6, 10, 14, 18, 22, ...    Choices: (A)a = a + a + 4 (B)a = 7/3a (C)a = a - 4 (D)a = a + 4

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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). Looking at the sequence: 2, 6, 10, 14, 18, 22, ..., we can see that the difference between consecutive terms is constant. 6 - 2 = 4, 10 - 6 = 4, 14 - 10 = 4, etc. This indicates that the sequence is arithmetic.Find Difference: Find the common difference of the sequence. To find the common difference, we subtract any term from the term that follows it. Using the first two terms: 6 - 2 = 4. The common difference (d) is 4.Identify Formula: Identify the recursive formula for the given arithmetic sequence. Since the sequence is arithmetic with a common difference of 4, the recursive formula will involve adding 4 to the previous term. The recursive formula should be of the form: a_n = a_(n-1) + d, where d is the common difference. Substituting 4 for d, we get: a_n = a_(n-1) + 4.Match with Choices: Match the recursive formula with the given choices. The correct recursive formula we found is a_n = a_(n-1) + 4. Comparing this with the given choices: (A) a = a + a + 4 (Incorrect, this does not make sense mathematically) (B) a = 7/3a (Incorrect, this suggests a geometric sequence) (C) a = a - 4 (Incorrect, this would decrease the sequence by 4 each time) (D) a = a + 4 (Correct, this matches our formula) Therefore, the correct choice is (D) a = a + 4.

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