# 7.1.1: Recursive Formulas

- Page ID
- 14782

## Describing the Pattern and Writing a Recursive Rule for a Sequence

In 2013, the days a full moon appeared were in the following **sequence** (with Jan. 1 being Day 1). Write a **recursive formula** for the sequence.

9, 38, 67, 96, ...

Source: Moongiant

## Recursive Rule

A **recursive rule** for a sequence is a formula which tells us how to progress from one term to the next in a sequence. Generally, the variable \(\ n\) is used to represent the term number. In other words, \(\ n\) takes on the values 1 (first term), 2 (second term), 3 (third term), etc. The variable, \(\ a_{n}\) represents the \(\ n^{t h}\) term and \(\ a_{n-1}\) represents the term preceding \(\ a_{n}\).

Example sequence: \(\ 4,7,11,16, \ldots, a_{n-1}, a_{n}\)

In the above sequence, \(\ a_{1}=4\), \(\ a_{2}=7\), \(\ a_{3}=11\) and \(\ a_{4}=16\).

Let's describe the pattern and write a recursive rule for the sequence: 9,11,13,15,…

First we need to determine what the pattern is in the sequence. If we subtract each term from the one following it, we see that there is a **common difference** of 29. We can therefore use \(\ a_{n-1}\) and \(\ a_{n}\) to write a recursive rule as follows: \(\ a_{n}=a_{n-1}+29\)

Now, let's write a recursive rule for the following sequences.

- 3,9,27,81,…
In this sequence, each term is multiplied by 3 to get the next term. We can write a recursive rule: \(\ a_{n}=3 a_{n-1}\)

- 1,1,2,3,5,8,…
This is a special sequence called the Fibonacci sequence. In this sequence each term is the sum of the previous two terms. We can write the recursive rule for this sequence as follows: \(\ a_{n}=a_{n-2}+a_{n-1}\).

## Examples

Earlier, you were asked to write a recursive formula for the sequence 9, 38, 67, 96, ...

**Solution**

First we need to determine what pattern the sequence is following. If we subtract each term from the one following it, we find that there is a common difference of 29. We can therefore use \(\ a_{n-1}\) and \(\ a_{n}\) to write a recursive rule as follows: \(\ a_{n}=a_{n-1}+29\)

**Write the recursive rules for the following sequences.**

1, 2, 4, 8, …

**Solution**

In this sequence each term is double the previous term so the recursive rule is: \(\ a_{n}=2 a_{n-1}\)

1, −2, −5, −8, …

**Solution**

This time three is subtracted each time to get the next term: \(\ a_{n}=a_{n-1}-3\)

1, 2, 4, 7, …

**Solution**

This one is a little trickier to express. Try looking at each term as shown below:

\(\ \begin{array}{l}

a_{1}&=1 \\

a_{2}&=a_{1}+1 \\

a_{3}&=a_{2}+2 \\

a_{4}&=a_{3}+3 \\

&\vdots \\

a_{n}&=a_{n-1}+(n-1)

\end{array}\)

## Review

Describe the pattern and write a recursive rule for the following sequences.

- \(\ \frac{1}{4},-\frac{1}{2}, 1,-2\) …
- 5, 11, 17, 23, …
- 33, 28, 23, 18, …
- 1, 4, 16, 64, …
- 21, 30, 39, 48, …
- 100, 75, 50, 25, …
- 243, 162, 108, 72, …
- 128, 96, 72, 54, …
- 1, 5, 10, 16, 23, …
- 0, 2, 2, 4, 6, …
- 3, 5, 8, 12, …
- 0, 2, 6, 12, …
- 4, 9, 14, 19, …
- \(\ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}\) …
- 4, 5, 9, 14, 23, …

## Answers for Review Problems

To see the Review answers, open this PDF file and look for section 11.2.

## Vocabulary

Term | Definition |
---|---|

common difference |
Every arithmetic sequence has a common or constant difference between consecutive terms. For example: In the sequence 5, 8, 11, 14..., the common difference is "3". |

common ratio |
Every geometric sequence has a common ratio, or a constant ratio between consecutive terms. For example in the sequence 2, 6, 18, 54..., the common ratio is 3. |

index |
The index of a term in a sequence is the term’s “place” in the sequence. |

recursive |
The recursive formula for a sequence allows you to find the value of the n^{th} term in the sequence if you know the value of the (n-1)^{th} term in the sequence. |

recursive formula |
The recursive formula for a sequence allows you to find the value of the n^{th} term in the sequence if you know the value of the (n-1)^{th} term in the sequence. |

sequence |
A sequence is an ordered list of numbers or objects. |