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Use the initial term and the recursive formula to find an explicit formula for the sequence aa. Write your answer in simplest form. \newlinea=−16a = -16 \newlinea=a+5a = a + 5 \newlinea = ______

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Q. Use the initial term and the recursive formula to find an explicit formula for the sequence aa. Write your answer in simplest form. \newlinea=−16a = -16 \newlinea=a+5a = a + 5 \newlinea = ______
  1. Identify Terms: Identify the initial term and the recursive formula.\newlineThe initial term is given as a1=−16a_1 = -16.\newlineThe recursive formula is given as an=an−1+5a_n = a_{n-1} + 5.
  2. Determine Sequence Type: Determine the type of sequence. The recursive formula an=an−1+5a_n = a_{n-1} + 5 suggests that we are dealing with an arithmetic sequence because the difference between consecutive terms is constant.
  3. Find Common Difference: Find the common difference in the arithmetic sequence.\newlineThe recursive formula an=an−1+5a_n = a_{n-1} + 5 indicates that the common difference dd is 55.
  4. Write Explicit Formula: Write the explicit formula for an arithmetic sequence.\newlineThe explicit formula for an arithmetic sequence is an=a1+d(n−1)a_n = a_1 + d(n - 1), where a1a_1 is the first term and dd is the common difference.
  5. Substitute Values: Substitute the known values into the explicit formula.\newlineSubstitute −16-16 for a1a_1 and 55 for dd into the formula an=a1+d(n−1)a_n = a_1 + d(n - 1) to find the explicit formula.\newlinean=−16+5(n−1)a_n = -16 + 5(n - 1)
  6. Simplify Formula: Simplify the explicit formula. \newlinean=−16+5n−5a_n = -16 + 5n - 5\newlinean=5n−21a_n = 5n - 21\newlineThe explicit formula for the sequence is an=5n−21a_n = 5n - 21.

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