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Which recursive formula can be used to define this sequence for n>1 n > 1 ?\newline15,13,11,9,7,5, 15, 13, 11, 9, 7, 5, \ldots \newlineChoices:\newline[A]an=an12[A] a_n = a_{n - 1} - 2 \newline[B]an=an1+an2+2[B] a_n = a_{n - 1} + a_{n - 2} + 2 \newline[C]an=1315an1[C] a_n = \frac{13}{15}a_{n - 1} \newline[D]an=12an1[D] a_n = -\frac{1}{2}a_{n - 1}

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Q. Which recursive formula can be used to define this sequence for n>1 n > 1 ?\newline15,13,11,9,7,5, 15, 13, 11, 9, 7, 5, \ldots \newlineChoices:\newline[A]an=an12[A] a_n = a_{n - 1} - 2 \newline[B]an=an1+an2+2[B] a_n = a_{n - 1} + a_{n - 2} + 2 \newline[C]an=1315an1[C] a_n = \frac{13}{15}a_{n - 1} \newline[D]an=12an1[D] a_n = -\frac{1}{2}a_{n - 1}
  1. Determine sequence type: We need to determine if the sequence is arithmetic, geometric, or neither. To do this, we look at the differences or ratios between terms.
  2. Calculate difference between first two terms: The sequence given is 15,13,11,9,7,5,15, 13, 11, 9, 7, 5, \ldots. We calculate the difference between the first two terms: 1315=213 - 15 = -2.
  3. Check consistency of difference: We check if this difference is consistent by calculating the difference between the next two terms: 1113=211 - 13 = -2.
  4. Conclude arithmetic sequence: Since the difference between each pair of consecutive terms is the same, we conclude that the sequence is arithmetic with a common difference of 2-2.
  5. Find recursive formula: Now we need to find a recursive formula that represents this arithmetic sequence. The recursive formula for an arithmetic sequence is generally given by an=an1+da_n = a_{n-1} + d, where dd is the common difference.
  6. Substitute common difference: Substituting the common difference we found into the recursive formula, we get an=an12a_n = a_{n-1} - 2.
  7. Compare with given choices: We compare our derived formula with the given choices. The correct recursive formula that matches our derived formula is an=an12a_n = a_{n-1} - 2.

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