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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=−35a_1 = -35\newlinean=an−1−2a_n = a_{n - 1} - 2\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=−35a_1 = -35\newlinean=an−1−2a_n = a_{n - 1} - 2\newlinean=_a_n = \_
  1. Initial Term: The initial term is a1=−35a_1 = -35.
  2. Recursive Formula: The recursive formula is an=an−1−2a_n = a_{n - 1} - 2. This means each term is 22 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=a1−2=−35−2=−37a_2 = a_1 - 2 = -35 - 2 = -37.
  5. Calculate a3a_3: For n=3n=3, a3=a2−2=−37−2=−39a_3 = a_2 - 2 = -37 - 2 = -39.
  6. Pattern Observation: We notice that each term is decreasing by 22 from the previous term, so the explicit formula will have the form an=a1−2(n−1)a_n = a_1 - 2(n - 1).
  7. Substitute a1a_1: Substitute a1=−35a_1 = -35 into the formula: an=−35−2(n−1)a_n = -35 - 2(n - 1).
  8. Simplify Formula: Simplify the formula: an=−35−2n+2a_n = -35 - 2n + 2.
  9. Combine Like Terms: Combine like terms: an=−2n−33an = -2n - 33.

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