Why is a sequence a function and how can you write a rule for a sequence?

Certain sequences (not all) can be defined (expressed) in a "recursive" form.

In a recursive formula, each term is defined as a function of its preceding term (s).

[Each term is found by doing something to the term (s) immediately in front of that term.]

A recursive formula designates the starting term, a 1, and the nth term of the sequence, an, as an expression containing the previous term (the term before it), an-1.

The process of recursion can be thought of as climbing a ladder.

To get to the third rung, you must step on the second rung. Each rung on the ladder depends upon stepping on the rung below it. You start on the first rung of the ladder. a1

From the first rung, you move to the second rung. a 2

a2 = a1 + "step up"

From the second rung, you move to the third rung. a3

a3 = a2 + "step up"

If you are on the n th rung, you must have stepped on the n-1st rung. an = a n-1 + "step up" Notation: Recursive forms work with the term (s) immediately in front of the term being examined. The table at the right shows that there are many options as to how this relationship may be expressed in notations. A recursive formula is written with two parts: a statement of the first term along with a statement of the formula relating successive terms. The statements below are all naming the same sequence: Given TermTerm in front

of Given Term a4a3 ana n-1 a n+1 an a n+4 a n+3 f (6) f (5) f (n) f (n-1) f (n+1) f (n)

Sequence: {10, 15, 20, 25, 30, 35, ... }. Find a recursive formula.

This example is an arithmetic sequence (the same number, 5, is added to each term to get to the next term).