The Next Term of a Recursively Described Sequence (F 502)

A recursive rule gives you two things: a starting value and an instruction for turning one term into the next. That instruction alone is useless — "a(n) = a(n-1) - 4" could describe infinitely many sequences depending on where you start. Pin down a(1), and suddenly every later term is forced. Change a(1) and the whole sequence shifts. That's why the starting value isn't optional extra information — it's half the definition.

To actually find a term, you unroll the rule: apply it once to get from a(1) to a(2), apply it again to get a(2) to a(3), and so on. Take a(n) = 2a(n-1) + 3 with a(1) = 4. Then a(2) = 2(4) + 3 = 11, a(3) = 2(11) + 3 = 25, a(4) = 2(25) + 3 = 53. Each term depends entirely on the one just before it, so skipping ahead isn't possible — you can't compute a(50) without passing through a(49), which needs a(48), all the way back to a(1).

This is what separates recursive definitions from explicit ones, and why explicit formulas matter: an explicit formula lets you plug in n and jump straight to any term, no unrolling required. Sometimes you can convert a recursive rule into an explicit one, especially for arithmetic sequences (constant difference) or geometric sequences (constant ratio), where the pattern of repeated addition or repeated multiplication has a clean closed form.