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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=31a_1 = 31\newlinean=an−1−16a_n = a_{n - 1} - 16\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=31a_1 = 31\newlinean=an−1−16a_n = a_{n - 1} - 16\newlinean=_a_n = \_
  1. Identify Sequence Type: Initial term is a1=31a_1 = 31, and the recursive formula is an=an−1−16a_n = a_{n - 1} - 16. This looks like an arithmetic sequence because we're subtracting the same number each time.
  2. Find Common Difference: The common difference dd in this sequence is −16-16 since that's what we subtract to get from one term to the next.
  3. Use Explicit Formula: The explicit formula for an arithmetic sequence is an=a1+d(n−1)a_n = a_1 + d(n - 1). We know a1=31a_1 = 31 and d=−16d = -16.
  4. Plug in Values: Plug the values into the formula: an=31−16(n−1)a_n = 31 - 16(n - 1).
  5. Simplify Formula: Simplify the formula: an=31−16n+16a_n = 31 - 16n + 16.
  6. Combine Like Terms: Combine like terms: an=−16n+47an = -16n + 47.
  7. Final Explicit Formula: So the explicit formula for the sequence is an=−16n+47a_n = -16n + 47.

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