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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=−29a_1 = -29\newlinean=an−1−19a_n = a_{n - 1} - 19\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=−29a_1 = -29\newlinean=an−1−19a_n = a_{n - 1} - 19\newlinean=_a_n = \_
  1. Initial Term: The initial term is a1=−29a_1 = -29.
  2. Recursive Formula: The recursive formula is an=an−1−19a_n = a_{n - 1} - 19. This means each term is 1919 less than the previous term.
  3. Find Explicit Formula: To find the explicit formula, we start with the first term and apply the recursive formula repeatedly to find a pattern.
  4. Calculate a2a^2: a2=a1−19=−29−19=−48a^2 = a^1 - 19 = -29 - 19 = -48.
  5. Calculate a3a_3: a3=a2−19=−48−19=−67a_3 = a_2 - 19 = -48 - 19 = -67.
  6. Identify Pattern: We notice that each term is 19n19n less than the first term when nn is the term number.
  7. Explicit Formula: So, the explicit formula is an=a1−19(n−1)a_n = a_1 - 19(n - 1).
  8. Substitute a1a_1: Substitute a1=−29a_1 = -29 into the formula: an=−29−19(n−1)a_n = -29 - 19(n - 1).
  9. Simplify Formula: Simplify the formula: an=−29−19n+19a_n = -29 - 19n + 19.
  10. Combine Like Terms: Combine like terms: an=−19n−10an = -19n - 10.

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