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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+2a_n = a_{n - 1} + 2\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+2a_n = a_{n - 1} + 2\newlinean=_a_n = \_
  1. Identify Pattern: Given a1=1a_1 = 1 and an=an−1+2a_n = a_{n - 1} + 2, let's find the first few terms to identify the pattern.\newlinea2=a1+2=1+2=3a_2 = a_1 + 2 = 1 + 2 = 3.\newlinea3=a2+2=3+2=5a_3 = a_2 + 2 = 3 + 2 = 5.\newlinea4=a3+2=5+2=7a_4 = a_3 + 2 = 5 + 2 = 7.
  2. Arithmetic Sequence: Notice the pattern; each term is 22 more than the previous term, which suggests an arithmetic sequence with a common difference of 22.
  3. 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.
  4. Substitute Values: Substitute a1=1a_1 = 1 and d=2d = 2 into the formula: an=1+(n−1)×2a_n = 1 + (n - 1) \times 2.
  5. Simplify Formula: Simplify the formula: an=1+2n−2a_n = 1 + 2n - 2.
  6. Combine Like Terms: Combine like terms: an=2n−1an = 2n - 1.

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