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Which recursive formula can be used to define this sequence for n>1n > 1?\newline5,18,31,44,57,70,5, 18, 31, 44, 57, 70, \ldots\newlineChoices:\newline(A)an=an1+an213a_n = a_{n-1} + a_{n-2} - 13\newline(B)an=an113a_n = a_{n-1} - 13\newline(C)an=185an1a_n = \frac{18}{5}a_{n-1}\newline(D)an=an1+13a_n = a_{n-1} + 13

Full solution

Q. Which recursive formula can be used to define this sequence for n>1n > 1?\newline5,18,31,44,57,70,5, 18, 31, 44, 57, 70, \ldots\newlineChoices:\newline(A)an=an1+an213a_n = a_{n-1} + a_{n-2} - 13\newline(B)an=an113a_n = a_{n-1} - 13\newline(C)an=185an1a_n = \frac{18}{5}a_{n-1}\newline(D)an=an1+13a_n = a_{n-1} + 13
  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 Difference: The first two terms are 55 and 1818. The difference between them is 185=1318 - 5 = 13.
  3. Check Consistency: We check if this difference is consistent by looking at the next pair of consecutive terms: 1818 and 3131. The difference is 3118=1331 - 18 = 13.
  4. Sequence Type Conclusion: Since the difference between consecutive terms is consistent, we can conclude that the sequence is arithmetic with a common difference of 1313.
  5. Find Recursive Formula: 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. Substitute Common Difference: Substituting the common difference of 1313 into the formula, we get the recursive formula an=an1+13a_n = a_{n-1} + 13.

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