Exercise 4.

Consider the sequence \(5, 11, 19, 29, 41, 55,\ldots\text{.}\) Assume \(a_1 = 5\text{.}\)

  1. Find a closed formula for \(a_n\text{,}\) the \(n\)th term of the sequence, by writing each term as a sum of a sequence. Hint: first find \(a_0\text{,}\) but ignore it when collapsing the sum.

  2. Find a closed formula again, this time using either polynomial fitting or the characteristic root technique (whichever is appropriate). Show your work.

  3. Find a closed formula once again, this time by recognizing the sequence as a modification to some well known sequence(s). Explain.

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