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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=−27a_1 = -27\newlinean=an−1+17a_n = a_{n - 1} + 17\newlinean=_____a_n = \_\_\_\_\_

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Full solution

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=−27a_1 = -27\newlinean=an−1+17a_n = a_{n - 1} + 17\newlinean=_____a_n = \_\_\_\_\_
  1. Initial Term: The initial term is a1=−27a_1 = -27.
  2. Recursive Formula: The recursive formula is an=an−1+17a_n = a_{n - 1} + 17. This means each term is 1717 more than the previous term.
  3. Explicit Formula: To find the explicit formula, we start with the first term and add 17(n−1)17(n - 1) to it, because the first term does not get added to itself, and each subsequent term adds 1717 once more than the previous term.
  4. Substitute Initial Term: So the explicit formula is an=a1+17(n−1)a_n = a_1 + 17(n - 1).
  5. Distribute 1717: Substitute a1=−27a_1 = -27 into the formula: an=−27+17(n−1)a_n = -27 + 17(n - 1).
  6. Combine Like Terms: Distribute the 1717: an=−27+17n−17an = -27 + 17n - 17.
  7. Combine Like Terms: Distribute the 1717: an=−27+17n−17an = -27 + 17n - 17.Combine like terms: an=17n−44an = 17n - 44.

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