Exercise 2.

For each sequence given below, find a closed formula for \(a_n\text{,}\) the \(n\)th term of the sequence (assume the first terms are \(a_0\)) by relating it to another sequence for which you already know the formula. In each case, briefly say how you got your answers.

  1. 4, 5, 7, 11, 19, 35, …

  2. 0, 3, 8, 15, 24, 35, …

  3. 6, 12, 20, 30, 42, …

  4. 0, 2, 7, 15, 26, 40, 57, … (Cryptic Hint: these might be called “house numbers”)

in-context