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Use the formulas \(T_n = \frac{n(n+1)}{2}\) and \(a_n = 2^n\) to find closed formulas that agree with the following sequences. Assume each first term corresponds to \(n=0\text{.}\)

  1. \((b_n)\text{:}\) \(1, 2, 4, 7, 11, 16, 22, \ldots \text{.}\)

  2. \((c_n)\text{:}\) \(3, 5, 9, 17, 33,\ldots \text{.}\)

  3. \((d_n)\text{:}\) \(0, 2, 6, 12, 20, 30, 42,\ldots \text{.}\)

  4. \((e_n)\text{:}\) \(3, 6, 10, 15, 21, 28, \ldots\text{.}\)

  5. \((f_n)\text{:}\) \(0, 1, 3, 7, 15, 31, \ldots \text{.}\)

  6. \((g_n)\) \(3, 6, 12, 24, 48, \ldots \text{.}\)

  7. \((h_n)\text{:}\) \(6, 10, 18, 34, 66, \ldots \text{.}\)

  8. \((j_n)\text{:}\) \(15, 33, 57, 87, 123, \ldots\text{.}\)

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