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Which recursive formula can be used to define this sequence for n>1n > 1?\newline\newline1,15,29,43,57,71,1, 15, 29, 43, 57, 71, \ldots\newline\newlineChoices:\newline(A) an=an114a_n = a_{n-1} - 14\newline(B) an=an1+14a_n = a_{n-1} + 14\newline(C) an=4315ana_n = \frac{43}{15}a_n\newline(D) an=an1+an114a_n = a_{n-1} + a_{n-1} - 14

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Q. Which recursive formula can be used to define this sequence for n>1n > 1?\newline\newline1,15,29,43,57,71,1, 15, 29, 43, 57, 71, \ldots\newline\newlineChoices:\newline(A) an=an114a_n = a_{n-1} - 14\newline(B) an=an1+14a_n = a_{n-1} + 14\newline(C) an=4315ana_n = \frac{43}{15}a_n\newline(D) an=an1+an114a_n = a_{n-1} + a_{n-1} - 14
  1. Determine Sequence Type: We need to determine if the sequence is arithmetic or geometric. To do this, we look at the difference between consecutive terms.
  2. Calculate Differences: The first two terms are 11 and 1515. The difference between them is 151=1415 - 1 = 14.
  3. Identify Arithmetic Sequence: The next two terms are 1515 and 2929. The difference between them is 2915=1429 - 15 = 14.
  4. Find Recursive Formula: Since the difference between consecutive terms is constant, we can conclude that the sequence is arithmetic with a common difference of 1414.
  5. Substitute Common Difference: To find the recursive formula for an arithmetic sequence, we use the formula an=an1+da_n = a_{n-1} + d, where dd is the common difference.
  6. Match with Correct Formula: Substituting the common difference of 1414 into the formula, we get an=an1+14a_n = a_{n-1} + 14.
  7. Match with Correct Formula: Substituting the common difference of 1414 into the formula, we get an=an1+14a_n = a_{n-1} + 14.Looking at the given choices, the correct recursive formula that matches our calculation is (B) an=an1+14a_n = a_{n-1} + 14.

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