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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=−47a_1 = -47\newlinean=an−1−7a_n = a_{n - 1} - 7\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=−47a_1 = -47\newlinean=an−1−7a_n = a_{n - 1} - 7\newlinean=_a_n = \_
  1. Initialize Term and Recursive Formula: The initial term is a1=−47a_1 = -47. The recursive formula is an=an−1−7a_n = a_{n - 1} - 7. To find the explicit formula, we need to express ana_n in terms of nn.
  2. First Few Terms Analysis: Let's look at the first few terms to see the pattern. a1=−47a_1 = -47, a2=a1−7=−47−7=−54a_2 = a_1 - 7 = -47 - 7 = -54, a3=a2−7=−54−7=−61a_3 = a_2 - 7 = -54 - 7 = -61.
  3. Identify Arithmetic Sequence Pattern: We notice that each term is 77 less than the previous term, which means the sequence is arithmetic with a common difference of −7-7.
  4. Explicit Formula for Arithmetic Sequence: The explicit formula for an arithmetic sequence is an=a1+(n−1)da_n = a_1 + (n - 1)d, where dd is the common difference.
  5. Substitute Values into Formula: Substitute a1=−47a_1 = -47 and d=−7d = -7 into the formula: an=−47+(n−1)(−7)a_n = -47 + (n - 1)(-7).
  6. Simplify Formula: Simplify the formula: an=−47−7n+7a_n = -47 - 7n + 7.
  7. Combine Like Terms: Combine like terms: an=−7n−40an = -7n - 40.

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