# Recursive formula

Here you will learn about recursive formulas, including what they are and how to write them.

Students will first learn about recursive formulas as part of functions in high school.

## What is a recursive formula?

The recursive formula is an equation that uses recursion to relate terms in a sequence. Recursion uses a rule over and over again. This relationship can be used to find the next term or the previous term.

For example,

Each number in a sequence is called a term, and each term is identified by its position within the sequence. For this sequence, the difference from term to term is common, making it an arithmetic sequence. You can write the first few terms as,

The first term, a_1=2

The second term, a_2=6

The third term, a_3=10

The fourth term, a_4=14

The fifth term, a_5=18

The n th term, a_n

The previous term, a_{n-1}

The next term, a_{n+1}

One way of generating this sequence would be to use a recursive formula, where each term is generated using the previous value.

When n=1, a_1=2.

When n=2, a_2=2+4=6.

When n=3, a_3=6+4=10, and so on.

So, the recursive formula is written as a_{n+1}=a_n+4.

The initial value, a_1, would need to be provided. The initial value could also be a_0.

Recursive formulas can be used to find particular solutions, but require starting at the beginning and repeatedly applying the rule to each term until the desired term is reached.

Another way of generating this sequence would be to use an explicit formula, which defines the relationship between the terms position and the value of the term itself.

To generate this sequence you can use the explicit formula for the sequence. For this sequence, the nth term would be a_n=2+4(n-1).

When n=1, a_1=2+4(0)=2.

When n=2, a_2=2+4(1)=6, and so on.

Step-by-step guide: Explicit formula

### Fibonacci sequence

The following sequence of numbers is a special type of recursive sequence called the Fibonacci sequence.

1, 1, 2, 3, 5, 8, 13, 21, …

The next number of the sequence can be found by adding the two previous terms together. Therefore, the Fibonacci numbers can be written using a recursive formula. You need to be given the first 2 numbers in the Fibonacci sequence.

F_{n+1}=F_{n-1}+F_n, \, F_1=1, \, F_2=1

If you change the first 2 values, you can generate a different Fibonacci sequence.

For example, F_{n+1}=F_{n-1}+F_n, \, F_1=2, \, F_2=5

would give the sequence 2, 5, 7, 12, 19, 31, …

In general, a Fibonacci sequence can be written as, F_{n+1}=F_{n-1}+F_n, \, F_1=a, \, F_2=b

and would give the sequence a, \, b, \, a+b, \, a+2b, \, 2a+3b, \, 3a+5b, …

Did you know the Fibonnaci numbers occur often in nature? Flowers often have 3 or 5 or 8 petals. This is why a clover with 4 petals is difficult to find and is considered lucky.

## Common Core State Standards

How does this relate to high school math?

• Functions – Building Functions (HSF.BF.A.1)
Write a function that describes a relationship between two quantities.

• Functions – Building Functions (HSF.BF.A.2)
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

• Functions – Linear, Quadratic and Exponential Models (HSF.LE.A.2)
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

## How to generate a sequence using a recursive formula

In order to generate a sequence using a recursive formula:

1. Find a recursive formula.
2. Substitute the given initial value into the formula to calculate the new value, \bf{\textbf{a}_{\textbf{n}+1}}.
3. Substitute the new value, \bf{\textbf{a}_{\textbf{n}+1}} , into the formula to calculate the next value.
4. Repeat step \bf{3} until the desired number of terms has been generated.

## Generating sequences using a recursive formula examples

### Example 1: generating an arithmetic sequence

A sequence is defined by a recursive formula a_{n+1}=a_n-4 and has a_0=100.

Find the next four terms of the sequence.

1. Find a recursive formula.

The recursive formula is given in the question, a_{n+1}=a_n-4.

2Substitute the given initial value into the formula to calculate the new value, \bf{\textbf{a}_{\textbf{n}+1}}.

The initial value is given in the question, a_0=100.

Substituting this into a recursive formula gives a_1=a_0-4=100-4=96.

3Substitute the new value, \bf{\textbf{a}_{\textbf{n}+1}} , into the formula to calculate the next value.

a_2=a_1-4=96-4=92

4Repeat step \bf{3} until the desired number of terms has been generated.

\begin{aligned} & a_3=a_2-4=92-4=88 \\\\ & a_4=a_3-4=88-4=84 \end{aligned}

The next four terms of the sequence are 96, 92, 88 and 84.

### Example 2: generating a geometric sequence

A sequence is defined by a recursive formula a_{n+1}=5 a_n and has a_0=1.

Find the next four terms of the sequence.

Find a recursive formula.

Substitute the given initial value into the formula to calculate the new value, \bf{\textbf{a}_{\textbf{n}+1}}.

Substitute the new value, \bf{\textbf{a}_{\textbf{n}+1}} , into the formula to calculate the next value.

Repeat step \bf{3} until the desired number of terms has been generated.

## How to find a recursive formula of a sequence

In order to find a recursive formula of a sequence:

1. Find the arithmetic or geometric relationship linking the terms.
2. Write a recursive formula with correct notation.
3. Give one term of the sequence as well as a recursive formula.

## Finding a recursive formula of a sequence examples

### Example 3: finding a recursive formula of an arithmetic sequence

Find a recursive formula that satisfies the sequence 5, 8, 11, 14, 17, …

Find the arithmetic or geometric relationship linking the terms.

Write a recursive formula with correct notation.

Give one term of the sequence as well as a recursive formula.

### Example 4: finding a recursive formula of a geometric sequence

Write down a recursive formula which produces the sequence 3, 6, 12, 24, 48, …

Find the arithmetic or geometric relationship linking the terms.

Write a recursive formula with correct notation.

Give one term of the sequence as well as a recursive formula.

### Example 5: finding a recursive formula of an arithmetic sequence

Describe the sequence 32, 37, 42, 47, … using a recursive formula.

Find the arithmetic or geometric relationship linking the terms.

Write a recursive formula with correct notation.

Give one term of the sequence as well as a recursive formula.

### Example 6: finding a recursive formula of a geometric sequence

Write down a recursive formula which produces the sequence 20, –10, 5, –2.5, 1.25, …

Find the arithmetic or geometric relationship linking the terms.

Write a recursive formula with correct notation.

Give one term of the sequence as well as a recursive formula.

### Teaching tips for recursive formula

• Give students time to explore the graphs and tables of each type of sequence to compare and contrast how the terms change as the sequence grows.

• Look for worksheets that include both arithmetic and geometric sequences. This requires students to practice identifying the type of sequence.

### Easy mistakes to make

• Confusing geometric and arithmetic sequences
An arithmetic sequence has a common difference from term to term. A geometric sequence has a common ratio from term to term.

For example,
3, 6, 9, 12, 15… is an arithmetic sequence.
3, 6, 12, 24, 48… is a geometric sequence.

• Confusing the explicit and recursive formulas for a sequence
The explicit formula shows how to find any term based on the relationships between the term number and the term itself. The recursive formula shows how to find the next term based on the previous term. It is easy to confuse the two.

For example,
3, 6, 9, 12, 15, …
The explicit formula is a_n=3+3(n-1).
The recursive formula is a_{n+1}=a_n+3.

• Confusing the notation
It can be easy to misunderstand the sequence notation, which uses subscripts to identify the terms of a sequence.

For example,
The first term of the sequence should always be defined, and is often a_1.
If a_1 is the first term, then a_2 is the second term, and so on with all successive terms. a_n is the n th term, a_{n+1} is the next term, and a_{n-1} is the preceding term.

So a sequence looks like,
a_1, a_2, \ldots, a_{n-1}, a_n, a_{n+1}, \ldots

• Thinking that all arithmetic sequences add
An arithmetic sequence that grows larger will have a positive difference. However, an arithmetic sequence that grows smaller will have a negative difference and be represented by subtraction.

For example,
15, 12, 9, 6, 3… is an arithmetic sequence with the recursive formula a_{n+1}=a_n-3.

### Practice recursive formula questions

1) A sequence is defined by the recursive formula a_{n+1}=a_n-4 and a_1=-2.

Find the next four terms of the sequence.

-2, 2, 6, 10

-6, -10, -14, -18

6, 4, 2, 0

-6, -12, -24, -48

Substitute the given initial value into the formula to calculate the new value,   a_{n+1}.

The initial value is given in the question, a_1=-2.

Substituting this into a recursive formula gives…

\begin{aligned} & a_2=a_1-4=-2-4=-6 \\\\ & a_3=a_2-4=-6-4=-10 \\\\ & a_4=a_3-4=-10-4=-14 \\\\ & a_5=a_4-4=-14-4=-18 \end{aligned}

2) A sequence is defined by the recursive formula a_{n+1}=\cfrac{1}{4} \, a_n and a_o=1000.

Find the next four terms of the sequence.

1000 \cfrac{1}{4} \, , \, 1000 \cfrac{1}{2} \, , \, 1000 \cfrac{3}{4} \, , \, 1001

4000, \, 1000, \, 250, \, 62.5

4000 \cfrac{1}{4} \, , \, 4000 \cfrac{1}{2} \, , \, 4000 \cfrac{3}{4} \, , \, 4001

250, \, 62.5, \, 15.625, \, 3.90625

The formula is given in the question, a_{n+1}=\cfrac{1}{4} \, a_n , where the initial value is a_o=1,000.

Substitute the given initial value into the formula to calculate the new value,   a_{n+1}.

\begin{aligned} & a_1=\cfrac{1}{4} \, a_0=\frac{1}{4} \cdot 1000=250 \\\\ & a_2=\cfrac{1}{4} \, a_1=\frac{1}{4} \cdot 250=62.5 \\\\ & a_3=\cfrac{1}{4} \, a_2=\frac{1}{4} \cdot 62.5=15.625 \\\\ & a_4=\cfrac{1}{4} \, a_3=\frac{1}{4} \cdot 15.625=3.90625 \end{aligned}

3) Describe the sequence 1, 6, 11, 16, 21, … using a recursive formula.

a_{n+1}=a_n+5, \, a_1=1

a_n=a_{n+1}+5

a_{n+1}=5 a_n, \, a_1=1

a_{n+1}=a_n+5

Each term in this sequence equals the one before it, plus 5.

The recursive formula can be written as a_{n+1}=a_n+5.

One term in the sequence (typically the first), as well as a recursive formula is required for the description.

a_{n+1}=a_n+5, \, a_1=1.

4) Describe the sequence 34, 30, 26, 22, 18, … using a recursive formula.

a_n=a_{n+1}-4

a_{n+1}=4 a_n

a_{n+1}=a_n+4, \, a_1=34

a_{n+1}=a_n-4, \, a_1=34

Each term in this sequence equals the one before it, minus 4.

The recursive formula can be written as a_{n+1}=a_n-4.

One term in the sequence (typically the first), as well as a recursive formula is required for the description.

a_{n+1}=a_n-4, \, a_1=34

5) Describe the sequence 4, 8, 16, 32, 64, … using a recursive formula.

a_n=2 a_{n+1}

a_{n+1}=2 a_{n}, \, a_1=4

a_{n+1}=2 a_n

a_{n+1}=\cfrac{1}{2} \, a_n, \, a_1=4

Each term in this sequence is double the previous term.

The recursive formula can be written as a_{n+1}=2 a_n.

One term in the sequence (typically the first), as well as a recursive formula is required for the description.

a_{n+1}=2 a_n, \, a_1=4

6) Describe the sequence -1, -1.5, -2.25, -3.375, -5.0625, … using a recursive formula.

a_n=-1.5 a_{n+1}

a_{n+1}=1.5 a_{n}, \, a_1=-1

a_{n+1}=-0.5+a_n

a_{n+1}=1.5 a_n

Each term in this sequence is 1.5 times the previous term.

The recursive formula can be written as a_{n+1}=1.5 a_n.

One term in the sequence (typically the first), as well as a recursive formula is required for the description.

a_{n+1}=1.5 a_{n}, \, a_1=-1

## Recursive formula FAQs

Does the first term have to be a whole number?

No, the first term can be an integer and can also involve fractions or decimals or any real number.

What is the recursive definition?

This is another name for the recursive formula, which defines how the sequence changes from term to term.

What is a recurrence relation?

A recurrence relation is an equation that identifies an initial condition (term before the other terms) and the rule for calculating the preceding terms. It is another name for the recursive formula of a recursive function.

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