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Use the initial term and the recursive formula to find an explicit formula for the sequence ana_n. Write your answer in simplest form.\newlinea1=19a_1 = 19\newlinean=an−1−9a_n = a_{n - 1} - 9\newlinean=_____a_n = \_\_\_\_\_

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Q. Use the initial term and the recursive formula to find an explicit formula for the sequence ana_n. Write your answer in simplest form.\newlinea1=19a_1 = 19\newlinean=an−1−9a_n = a_{n - 1} - 9\newlinean=_____a_n = \_\_\_\_\_
  1. Initial Term: The initial term is a1=19a_1 = 19.
  2. Recursive Formula: The recursive formula is an=an−1−9a_n = a_{n - 1} - 9. This means each term is 99 less than the previous term.
  3. Explicit Formula: To find the explicit formula, we start with the initial term and apply the recursive formula repeatedly.
  4. Calculate a2a_2: For n=2n=2, a2=a1−9=19−9=10a_2 = a_1 - 9 = 19 - 9 = 10.
  5. Calculate a3a_3: For n=3n=3, a3=a2−9=10−9=1a_3 = a_2 - 9 = 10 - 9 = 1.
  6. Pattern Recognition: We notice that each term is reduced by 99 from the previous term, so the explicit formula will have the form an=a1−9(n−1)a_n = a_1 - 9(n - 1).
  7. Substitute a1a_1: Substitute a1=19a_1 = 19 into the formula: an=19−9(n−1)a_n = 19 - 9(n - 1).
  8. Simplify Formula: Simplify the formula: an=19−9n+9a_n = 19 - 9n + 9.
  9. Combine Like Terms: Combine like terms: an=28−9nan = 28 - 9n.

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