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# Recursive Formulas

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Describing the Pattern and Writing a Recursive Rule for a Sequence
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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, ...

### 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 $n$ 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 $n^{th}$ 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 $a_4=16$ .

#### Example A

Describe the pattern and write a recursive rule for the sequence: $9, 11, 13, 15, \ldots$

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 $a_{n-1}$ and $a_n$ to write a recursive rule as follows: $a_n=a_{n-1}+29$

#### Example B

Write a recursive rule for the sequence: $3, 9, 27, 81, \ldots$

Solution: In this sequence, each term is multiplied by 3 to get the next term. We can write a recursive rule: $a_n=3a_{n-1}$

#### Example C

Write a recursive rule for the sequence: $1, 1, 2, 3, 5, 8, \ldots$

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: $a_n=a_{n-2}+a_{n-1}$ .

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 $a_{n-1}$ and $a_n$ to write a recursive rule as follows: $a_n=a_{n-1}+29$

### Guided Practice

Write the recursive rules for the following sequences.

1. $1, 2, 4, 8, \ldots$

2. $1, -2, -5, -8, \ldots$

3. $1, 2, 4, 7, \ldots$

1. In this sequence each term is double the previous term so the recursive rule is: $a_n=2a_{n-1}$

2. This time three is subtracted each time to get the next term: $a_n=a_{n-1}-3$ .

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

$a_1 &=1 \\a_2 &=a_1+1 \\a_3 &=a_2+2 \\a_4 &=a_3+3 \\& \ \ \vdots \\a_n &=a_{n-1}+(n-1)$

### 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.

1. $\frac{1}{4}, -\frac{1}{2}, 1, -2 \ldots$
2. $5, 11, 17, 23, \ldots$
3. $33, 28, 23, 18, \ldots$
4. $1, 4, 16, 64, \ldots$
5. $21, 30, 39, 48, \ldots$
6. $100, 75, 50, 25, \ldots$
7. $243, 162, 108, 72, \ldots$
8. $128, 96, 72, 54, \ldots$
9. $1, 5, 10, 16, 23, \ldots$
10. $0, 2, 2, 4, 6, \ldots$
11. $3, 5, 8, 12, \ldots$
12. $0, 2, 6, 12, \ldots$
13. $4, 9, 14, 19, \ldots$
14. $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5} \ldots$
15. $4, 5, 9, 14, 23, \ldots$