Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

4.17.2 What Is a Recursive Definition?

4.17.2 • What Is a Recursive Definition?

Activity

The recursive definition is a function that provides a repeated, or recurring, process to find terms in a sequence.

Start with the recursive definition: $R (n) = 3 R (n - 1) + 1$ , $n \geq 2$ and given $R (1) = - 1$ .

The first part of the recursive definition provides the instructions for how to generate the sequence. It provides the function rule or formula for the $n^{th}$ term.
The second part states the domain of the sequence. It describes where we start and the values that are substituted into the function.
The third part is the basis for the sequence or the smallest term value.

See the table below for how to substitute to find more terms in the sequence using the recursive definition. Substitute the term number, $n$ , in for $n$ in the recursive definition. Complete the table with a partner.

Term number (n)	Substitute in $R (n)$	Value
1	given	-1
2	$3 R (2 - 1) + 1 = 3 R (1) + 1 = 3 (- 1) + 1 = - 2$	-2
3	$3 R (3 - 1) + 1 = 3 R (2) + 1 = 3 (- 2) + 1 = - 5$	-5
4	$3 R (4 - 1) + 1 = 3 R (3) + 1 = 3 (- 5) + 1$	a. ___
5	a. ___	b. ___
6	b. ___	c. ___

1. Using the recursive definition:

a. Substitute $n = 5$ in $R (n)$

Compare your answer:

$3 R (5 - 1) + 1$

b. Substitute $n = 6$ in $R (n)$

Compare your answer:

$3 R (6 - 1) + 1$

2. Determine the value for each term:

a. Value when $n = 4$

Compare your answer:

–14

b. Value when $n = 5$

Compare your answer:

–41

c. Value when $n = 6$

Compare your answer:

–122

3. Write the full sequence out to 6 terms.

Compare your answer:

–1, –2, –5, –14, –41, –122

4. Determine if the following sequences are arithmetic or geometric.

a. 3, 6, 12, 24

Geometric

b. 18, 36, 72, 144

Geometric

c. 3, 8, 13, 18

Arithmetic

d. 18, 13, 8, 3

Arithmetic

e. 18, 9, 4.5, 2.25

Geometric

f. 18, 20, 22, 24

Arithmetic

g. 3, 15, 75, 375

Geometric

Classroom Connection

Desmos Activity:

Log into student.desmos.com using the information provided by your teacher to complete the activity.

5. Examine the following recursive definitions.

$F (1) = 18$ , $F (n) = F (n - 1) + 2$ , $n \geq 2$

$G (1) = 18$ , $G (n) = \frac{1}{2} \cdot G (n - 1)$ , $n \geq 2$

$H (1) = 3$ , $H (n) = 5 \cdot H (n - 1)$ , $n \geq 2$

$J (1) = 3$ , $J (n) = J (n - 1) + 5$ , $n \geq 2$

$K (1) = 18$ , $K (n) = K (n - 1) - 5$ , $n \geq 2$

$L (1) = 18$ , $L (n) = 2 \cdot L (n - 1)$ , $n \geq 2$

$M (1) = 3$ , $M (n) = 2 \cdot M (n - 1)$ , $n \geq 2$

Match each definition with one of the sequences from the previous question. Be prepared to explain to a partner how you know it's a match. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

a. 3, 6, 12, 24

$M (1) = 3$ , $M (n) = 2 \cdot M (n - 1)$ , $n \geq 2$

b. 18, 36, 72, 144

$L (1) = 18$ , $L (n) = 2 \cdot L (n - 1)$ , $n \geq 2$

c. 3, 8, 13, 18

$J (1) = 3$ , $J (n) = J (n - 1) + 5$ , $n \geq 2$

d. 18, 13, 8, 3

$K (1) = 18$ , $K (n) = K (n - 1) - 5$ , $n \geq 2$

e. 18, 9, 4.5, 2.25

$G (1) = 18$ , $G (n) = \frac{1}{2} \cdot G (n - 1)$ , $n \geq 2$

f. 18, 20, 22, 24

$F (1) = 18$ , $F (n) = F (n - 1) + 2$ , $n \geq 2$

g. 3, 15, 75, 375

$H (1) = 3$ , $H (n) = 5 \cdot H (n - 1)$ , $n \geq 2$

Video: Learning About Recursive Definitions and Sequences

Watch the following video to learn more about matching sequences to their recursive definitions.

Learning About Recursive Definitions and Sequences

Self Check

Which of the following sequences matches the recursive definition for $G(n)$ where $G(n)=G(n-1)+4$ for $n\geq2$ when $G(1)=3$ ?

3, 12, 48, 144
3, 7, 11, 15
1, 3, 7, 11, 15
4, 8, 12, 16, 20

Additional Resources

Writing Recursive Definitions

Sometimes we can define a sequence recursively. That is, we can describe how to calculate the next term in a sequence if we know the previous term.

Here’s a sequence: 6, 10, 14, 18, 22, . . . This is an arithmetic sequence, where each term is 4 more than the previous term. Since sequences are functions, let’s call this sequence $f$ . Then we can use function notation to write $f (n) = f (n - 1) + 4$ . Here, $f (n)$ is the term, $f (n - 1)$ is the previous term, and + 4 represents the common difference since $f$ is an arithmetic sequence.

When we define a function recursively, we must also say what the first term is. Without that, there would be no way of knowing if the sequence defined by $f (n) = f (n - 1) + 4$ started with 6 or 81 or any other number. Here, one possible initial condition is $f (1) = 6$ . It could also make sense to number the terms starting with 0, using $f (0) = 6$ .

Combining this information gives the recursive definition: $f (1) = 6$ and $f (n) = f (n - 1) + 4$ for $n \geq 2$ , where $n$ is an integer. We include the $n \geq 2$ at the end since the value of $f$ at 1 is already given, and the other terms in the sequence are generated by inputting integers larger than 1 into the definition.

Let’s look at some examples from the first activity and write the recursive definition for each type of sequence:

Example 1

Sequence A: $30, 40, 50, 60, 70, \dots$

Step 1 - Determine the type of sequence

This sequence is arithmetic.

Step 2 - Determine the common difference or common ratio

The terms increase by adding a value of 10, so the common difference is 10.

Step 3 - Determine the formula or rule

Since one term’s value comes from adding 10 to the term value before it, that means $n = (n - 1) + 10$ . Now write this rule using function notation.

$A (n) = A (n - 1) + 10$

Step 4 - Determine the initial condition or basis

$A (1) = 30$

Step 5 - Determine the domain of the formula

We use the first term to find the second term, so that is where we start at $n = 2$ . So, our domain is $n \geq 2$ .

The recursive definition for 30, 40, 50, 60, 70, … is $A (1) = 30$ , $A (n) = A (n - 1) + 10$ , $n \geq 2$ .

Example 2

Sequence B: $80, 40, 20, 10, 5, 2.5, \dots$

Step 1 - Determine the type of sequence

This sequence is geometric.

Step 2 - Determine the common difference or common ratio

The terms are divided by 2 each time, so the common ratio is $\frac{1}{2}$ (since it is divided by 2).

Step 3 - Determine the formula or rule

Since one term’s value comes from multiplying $\frac{1}{2}$ to the term value before it, that means $n = (n - 1) \times \frac{1}{2}$ . Now write this rule using function notation.

$B (n) = B (n - 1) \times \frac{1}{2}$

Step 4 - Determine the initial condition or basis

$B (1) = 80$

Step 5 - Determine the domain of the formula

We use the first term to find the second term, so that is where we start at $n = 2$ . So, our domain is $n \geq 2$ .

The recursive definition for 80, 40, 20, 10, 5, 2.5, … is $B (1) = 80$ , $B (n) = B (n - 1) \times \frac{1}{2}$ , $n \geq 2$ .

Example 3

Sequence F: $1, 3, 7, 15, 31, \dots$

Step 1 - Determine the type of sequence

This sequence is neither arithmetic nor geometric.

Try it

Writing Recursive Definitions

Write $E (1)$ and a recursive definition for $E (n)$ :

E: 20, 13, 6, –1, –8, . . .

Here is how to write the recursive definition for this sequence:

$E (1) = 20$ , then each term after is formed by subtracting 7, so $E (n) = E (n - 1) - 7$ , $n \geq 2$ .