A rule that looks back

You are told: "Add 3 to the term before." Where does the list start?

The rule

The rule alone does not say where the list starts. A first term must be given.

A recursive rule builds each term from the term before it.

Write tₙ for the term at position n. Then tₙ₋₁ is the term just before tₙ.

"Add 3 to the term before" is not complete. It needs a first term, or no term can be built.

"t₁ = 2, and tₙ = tₙ₋₁ + 3" is complete. It gives 2, 5, 8, 11.

A rule can do more than add. Take t₁ = 3. Double the term before, then take 1 away.

3 × 2 − 1 = 5. 5 × 2 − 1 = 9. 9 × 2 − 1 = 17. The list is 3, 5, 9, 17.