Sequence: {10, 15, 20, 25, 30, 35, ...}. Find a recursive formula. We saw how "explicit" arithmetic sequences are discrete linear functions. Since we are discussing the same sequences again, the "recursive" forms will also be discrete linear functions.

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 Formula for this example: in subscript notation: a1 = 10; an = an - 1+ 5 in function notation: f (1) = 10; f (n)= f (n - 1)+ 5	The graph and the list of terms can both verify that each term is increasing by a value of 5. The sequence starts at 10. This pattern of repetitive addition, allows calculation of terms based on preceding values. As a formula, we can easily state the first term to be 10, and then add 5 to each previous term.

In most arithmetic sequences, however, 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.

Remember that recursive formulas, unlike explicit formulas, will only let you find a term
in a sequence if you already know the term directly before the term you seek.

Origins of the Arithmetic Recursive Formula

The origin of the explicit formula for an arithmetic sequence was based upon the sequence pattern listed in the definition of an arithmetic sequence. The origin of the recursive formula for an arithmetic sequence is base upon another way of looking at that pattern.

(explicit)

(recursive)

If you start with the explicit pattern, and list out all of the "d"s in each term, you can see how the recursive pattern was developed by grouping within the pattern. The recursive formula shows that given a1, each term can be expressed as the preceding term plus "d ".

Recursive Formula of an arithmetic sequence:  Given a1 and an= an - 1 + d where a1 is the first term of the sequence, d is the common difference, n is the number of the term to find.

There are numerous ways to work with arithmetic sequences and recursive formulas. Let's take a look at some of the situations.

Question	Answer
1. Find the first 5 terms of this recursive arithmetic sequence:         a1 = 15         an = an - 1 + 3	1.  The formula shows that each term is three more than the term in front of it. This one is easy to see. 15, 18, 21, 24, 27, ...
2. Find the first 5 terms of this recursive arithmetic sequence.         a1 = 20         an = an - 1 - 12	2. The formula indicates that each term is 12 less than the term in front of it. Another easy one. 20. 8, -4, -16, -28, ...
3. Write a recursive formula for this arithmetic sequence.         -3, 17, 37, 57, ...	3.The first term is -3. The additional terms are 20 more than the previous term. a1 = -3 an = an - 1 + 20 The formula is easy if you can quickly see the pattern in the sequence.
4.  For the arithmetic sequence:         1, ½, 0, -½, ... a) Find the next three term. b) Write a recursive formula for this sequence.	4. a) 1, ½, 0, -½, -1, -1½, -2 b) a1 = 1 an = an - 1 - ½
5.  Find the first four terms of the arithmetic sequence         a1 = 10         an = an - 1 + 1.5	5. The first four term:  10, 11.5, 13, 14.5
6.  The first term of an arithmetic sequence is -4 and the third term is 36. Write a recursive formula to express this sequence.	6.  Get a visual of the situation: -4, ___, 36 In this arithmetic sequence the middle term (the second term) will be half way between the surrounding two terms. -4 + 36 = 32 ½ of 32 = 16 The missing term is 16, making the common difference to be 20. an = an - 1 + 20
7. a) Write a recursive formula for the sequence: {18, 21, 24, 27, ...}. b) Using the formula from part a, find the 100th term of this sequence.	a) We know a1= 18 and d = 4. a1 = 18  and   an = an - 1 + 4 b) WHOA!! Not doing this one!! To find this answer using a recursive formula, would require finding ALL of the terms prior to the 100th term. We are going to "cheat" on this one. The explicit formula page told us that the answer was 414.

To summarize the process of writing a recursive formula for an arithmetic sequence: 1. Determine if the sequence is arithmetic (Do you add, or subtract, the same amount from one term to the next?) 2. Find the common difference. (The number you add or subtract.) 3. Create a recursive formula by stating the first term, and then stating the formula to be the previous term plus the common difference. 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