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Use the explicit formula to find a recursive formula for the sequence ana_n. Write your answer in simplest form.\newlineThe recursive formula should depend on an−1a_{n - 1}.\newlinean=19n+14a_n = 19n + 14\newlinean=_a_n = \_

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Q. Use the explicit formula to find a recursive formula for the sequence ana_n. Write your answer in simplest form.\newlineThe recursive formula should depend on an−1a_{n - 1}.\newlinean=19n+14a_n = 19n + 14\newlinean=_a_n = \_
  1. Given Explicit Formula: The explicit formula is an=19n+14a_n = 19n + 14. Let's find the first few terms by plugging in n=1,2,3n = 1, 2, 3.
  2. Calculate First Few Terms: For n=1n = 1, a1=19(1)+14=33a_1 = 19(1) + 14 = 33.\newlineFor n=2n = 2, a2=19(2)+14=52a_2 = 19(2) + 14 = 52.\newlineFor n=3n = 3, a3=19(3)+14=71a_3 = 19(3) + 14 = 71.
  3. Identify Common Difference: Notice that the difference between consecutive terms is 1919. This is because when we increase nn by 11, we add 1919 to the previous term.
  4. Recursive Formula: The recursive formula for an arithmetic sequence is an=an−1+da_n = a_{n-1} + d, where dd is the common difference. Here, d=19d = 19.
  5. Calculate a1a_1: So the recursive formula is an=an−1+19a_n = a_{n-1} + 19. But we need to find a1a_1 to complete the formula.
  6. Complete Recursive Formula: We already calculated a1=33a_1 = 33. So the complete recursive formula is an=an−1+19a_n = a_{n-1} + 19, with a1=33a_1 = 33.

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