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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=−1a_1 = -1\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=−1a_1 = -1\newlinean=an−1+7a_n = a_{n - 1} + 7\newlinean=_____a_n = \_\_\_\_\_
  1. Initial Term and Recursive Formula: The initial term is a1=−1a_1 = -1. 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. Pattern of Terms: Let's look at the first few terms to see the pattern. a1=−1a_1 = -1, a2=a1+7=−1+7=6a_2 = a_1 + 7 = -1 + 7 = 6, a3=a2+7=6+7=13a_3 = a_2 + 7 = 6 + 7 = 13, and so on.
  3. Arithmetic Sequence: We notice that each term is 77 more than the previous term, which means the sequence is arithmetic with a common difference of 77.
  4. Explicit Formula: 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. Here, a1=−1a_1 = -1 and d=7d = 7.
  5. Substitute Values: Substitute the values into the formula: an=−1+(n−1)×7a_n = -1 + (n - 1) \times 7.
  6. Simplify Formula: Simplify the formula: an=−1+7n−7a_n = -1 + 7n - 7.
  7. Combine Like Terms: Combine like terms: an=7n−8an = 7n - 8.

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