Question

How do you write a recursive formula

Answer 1

You need to show how a term of formula is based on a previous term.

For example, let's say you have the sequence 3, 6, 9, 12, ...
(multiples of 3).

As a regular function, you can have y = 3x, for the domain of all integers greater than 0. Since the domain is 1, 2, 3, 4, ..., when you multiply those numbers by 3, to get y, you generate the sequence.

To define the sequence recursively, you must base each new number on the previous number.

Start by stating the first term.

a1 = 3

If you look at the sequence, 3, 6, 9, ..., you see it has a constant difference, and it is an arithmetic sequence, so to generate a term, you need to add 3 to the previous term.

Once we defined the first term to be 3, the next term is a2 = 3 + 3 = 6
Then a3 = 6 + 3 = 9, etc.

To define the sequence recursively, just generalize the addition of 3 to a term to get the next term.

$a_{1} = 3 a_{n} = 3 a_{n - 1}$