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

9, 38, 67, 96, ...

[Source: http://www.moongiant.com/Full_Moon_New_Moon_Calendar.php ]

### Guidance

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
is used to represent the term number. In other words,
takes on the values 1 (first term), 2 (second term), 3 (third term), etc. The variable,
represents the
term and
represents the term preceding
.

Example sequence:

In the above sequence, , , and .

#### Example A

Describe the pattern and write a recursive rule for the sequence:

**
Solution:
**
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
and
to write a recursive rule as follows:

#### Example B

Write a recursive rule for the sequence:

**
Solution:
**
In this sequence, each term is multiplied by 3 to get the next term. We can write a recursive rule:

#### Example C

Write a recursive rule for the sequence:

**
Solution:
**
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:
.

**
Intro Problem Revisit
**
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
and
to write a recursive rule as follows:

### Guided Practice

Write the recursive rules for the following sequences.

1.

2.

3.

#### Answers

1. In this sequence each term is double the previous term so the recursive rule is:

2. This time three is subtracted each time to get the next term: .

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

### Vocabulary

- Recursive Rule
- A rule that can be used to calculate a term in a sequence given the previous term(s).

### Explore More

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