4.17.2 What Is a Recursive Definition?

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 $n\ge 2$, where $n$ is an integer. We include the $n\ge 2$ 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 $n\ge 2$.

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

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 $n\ge 2$.

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

Example 3

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

Step 1 - Determine the type of sequence

This sequence is neither arithmetic nor geometric.