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Which recursive formula can be used to define this sequence for n>1n > 1?\newline10,13,16,19,22,25,10, 13, 16, 19, 22, 25, \ldots\newlineChoices:\newline(A)an=an1+3a_{n} = a_{n-1} + 3\newline(B)an=an1+an23a_{n} = a_{n-1} + a_{n-2} - 3\newline(C)an=1310an1a_{n} = \frac{13}{10}a_{n-1}\newline(D)an=an13a_{n} = a_{n-1} - 3

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Q. Which recursive formula can be used to define this sequence for n>1n > 1?\newline10,13,16,19,22,25,10, 13, 16, 19, 22, 25, \ldots\newlineChoices:\newline(A)an=an1+3a_{n} = a_{n-1} + 3\newline(B)an=an1+an23a_{n} = a_{n-1} + a_{n-2} - 3\newline(C)an=1310an1a_{n} = \frac{13}{10}a_{n-1}\newline(D)an=an13a_{n} = a_{n-1} - 3
  1. Determine Sequence Type: Determine if the sequence is arithmetic or geometric.\newlineWe 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).\newlineLooking at the sequence: 10,13,16,19,22,25,10, 13, 16, 19, 22, 25, \ldots\newlineThe difference between consecutive terms is:\newline1310=313 - 10 = 3\newline1613=316 - 13 = 3\newline1916=319 - 16 = 3\newline2219=322 - 19 = 3\newline2522=325 - 22 = 3\newlineSince the difference is constant, the sequence is arithmetic.
  2. Find Common Difference: Find the common difference of the arithmetic sequence.\newlineThe common difference dd is the difference between any two consecutive terms.\newlineUsing the first two terms:\newline1310=313 - 10 = 3\newlineTherefore, the common difference is 33.
  3. Identify Recursive Formula: Identify the recursive formula for the arithmetic sequence.\newlineFor an arithmetic sequence, the recursive formula is generally of the form:\newlinean=an1+da_n = a_{n-1} + d\newlineWhere ana_n is the nth term, an1a_{n-1} is the (n1)(n-1)th term, and dd is the common difference.\newlineSince we have determined that d=3d = 3, the recursive formula is:\newlinean=an1+3a_n = a_{n-1} + 3

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