Math 1 · Unit 9 · Lesson 4 · part 2 of 5

From Now-Next to formal notation

You have met two ways to describe a list.

From Now-Next to formal notation

You have met two ways to describe a list. A recursive rule says how to get the next term from the one before, like climbing stairs. An explicit rule says how to jump straight to any term, like taking the elevator to floor 50. This lesson is about changing one into the other.

Start with plain words. Many people first write a recursive rule as Now-Next:

Next = Now + 12, starting at 4

That gives 4, 16, 28, 40, ... It is easy to read, and it is a fine way to think. The formal notation says the same sentence with subscripts:

a_{n + 1} = a_n + 12, a₁ = 4

Say it: a sub n plus 1 equals a sub n, plus 12, and a sub 1 equals 4. Here a_n is Now and a_{n + 1} is Next. The second part, a₁ = 4, is the initial term: the starting point every recursive rule needs.

Two notations for the same thing
What it is	Function notation	Sequence notation	Say it
the term	f(n)	a_n	a sub n
the previous term	f(n − 1)	a_{n − 1}	a sub n minus 1
the next term	f(n + 1)	a_{n + 1}	a sub n plus 1
the first term	f(1)	a₁	a sub 1

Two ways to write the same recursive rule. a_{n + 1} = a_n + 12 (next from now) and a_n = a_{n − 1} + 12 (now from previous) mean the same thing. Both say: each term is 12 more than the one before. Books use both, so get comfortable reading both.

Where to go next

Next: Translating, in both directionsRecursive to explicit , in three steps. All the partsRecursive and Explicit, Both Ways