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  1. \(a_0 = 0\text{,}\) \(a_1 = 1\text{,}\) \(a_2 = 3\text{,}\) \(a_3 = 6\) \(a_4 = 10\text{.}\) The sequence was described by a closed formula. These are the triangular numbers. A recursive definition is: \(a_n = a_{n-1} + n\) with \(a_0 = 0\text{.}\)

  2. This is a recursive definition. We continue \(a_2 = 2\text{,}\) \(a_3 = 3\text{,}\) \(a_4 = 4\text{,}\) \(a_5 = 5\text{,}\) and so on. A closed formula is \(a_n = n\text{.}\)

  3. We have \(a_0 = 1\text{,}\) \(a_1 = 1\text{,}\) \(a_2 = 2\text{,}\) \(a_3 = 6\text{,}\) \(a_4 = 24\text{,}\) \(a_5 = 120\text{,}\) and so on. The closed formula is \(a_n = n!\text{.}\)

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