How to write a recursive formula for an arithmetic sequence

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. a2
    a2 = a1 + "step up"
From the second rung, you move to the third rung. a3
    a3 = a2 + "step up"

If you are on the nth rung, you must have stepped on the n-1st rung.    an = an-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:

Term Number	Term	Subscript Notation	Function Notation
1	10	a1	f (1)
2	15	a2	f (2)
3	20	a3	f (3)
4	25	a4	f (4)
5	30	a5	f (5)
6	35	a6	f (6)
n		an	f (n)

Recursive Formulas:
in subscript notation: a1 = 10; an = an-1+ 5
in function notation: f (1) = 10; f (n)= f (n-1)+ 5

Arithmetic sequences are linear in nature. Remember that the domain consists of the natural numbers, {1, 2, 3, ...}, and the range consists of the terms of the sequence. It may be the case with arithmetic sequences that the graph will increase or decrease.

In most arithmetic sequences, a recursive formula is easier to create than an explicit formula. The common difference is usually easily seen, which is then used to quickly create the recursive formula.

a1 = first term;
an= an-1 + d

a1 = the first term in the sequence
an = the nth term in the sequence
an-1 = the term before the nth term
n = the term number
d = the common difference.

{10, 15, 20, 25, 30, 35, ...}

first term = 10, common difference = 5
recursive formula: a1= 10; an= an-1 + 5

Term Number	Term	Subscript Notation	Function Notation
1	3	a1	f (1)
2	6	a2	f (2)
3	12	a3	f (3)
4	24	a4	f (4)
5	48	a5	f (5)
6	96	a6	f (6)
n		an	f (n)

Recursive Formulas:
in subscript notation: a1 = 3; an = 2 • an-1
in function notation: f (1) = 3; f (n) = 2 • f (n-1)

Notice that this sequence has an exponential appearance.
It may be the case with geometric sequences that the graph will increase or decrease.

a1 = first term;
an= r • an-1

a1 = the first term in the sequence
an = the nth term in the sequence
an-1 = the term before the nth term
n = the term number
r = the common ratio

{3, 6, 12, 24, 48, 96, ...}

first term = 3, common ratio = 2
explicit formula: an= 3 • 2n-1

While we have seen recursive formulas for arithmetic sequences and geometric sequences, there are also recursive forms for sequences that do not fall into either of these categories.

The sequence shown in this example is a famous sequence called the Fibonacci sequence.

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