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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=45a_1 = 45\newlinean=an−1+8a_n = a_{n - 1} + 8\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=45a_1 = 45\newlinean=an−1+8a_n = a_{n - 1} + 8\newlinean=_____a_n = \_\_\_\_\_
  1. Identify Initial Term and Recursive Formula: Initial term is a1=45a_1 = 45, and the recursive formula is an=an−1+8a_n = a_{n - 1} + 8. This looks like an arithmetic sequence with a common difference.
  2. Find Common Difference: The common difference dd can be found by looking at the recursive formula. Since an=an−1+8a_n = a_{n - 1} + 8, the common difference d=8d = 8.
  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 plug in a1=45a_1 = 45 and d=8d = 8 to get the explicit formula.
  4. Substitute Values: Substitute the values into the formula: an=45+8(n−1)a_n = 45 + 8(n - 1).
  5. Simplify Formula: Simplify the formula: an=45+8n−8a_n = 45 + 8n - 8.
  6. Combine Like Terms: Combine like terms: an=8n+37an = 8n + 37.
  7. Check Formula: Check the formula by plugging in n=1n=1 to see if we get the first term: a1=8(1)+37=45a_1 = 8(1) + 37 = 45. It matches the given initial term, so it seems right.

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