Which rule is easier for finding a far-away term?

An explicit rule is usually convenient when you want one term far into the sequence because the term number can be substituted directly.

Can the same sequence have both an explicit and a recursive rule?

Yes. The same sequence can often be described in both ways.

MATHEMATICS • SEQUENCES

Explicit and Recursive Rules in Sequences

Q: What is an explicit rule?

An explicit rule gives the nth term directly using the position n. You can substitute the term number without calculating all the earlier terms.

Q: What is a recursive rule?

A recursive rule gives a starting term and a rule for finding each new term from one or more previous terms.

Two rules can describe the same sequence. The trick is simply to understand what each rule is telling you: explicit means go directly to the term, while recursive means use the previous term to get the next one.

First, what is a sequence?

A sequence is an ordered list of numbers. Each number has a position, called a term number.

Here, 5 is the first term, 8 is the second term, 11 is the third term, and so on.

The big idea: two different ways to describe a sequence

⚡ Explicit rule

Jump directly to the term you want.

You use the term number n in a formula.

Think: “Tell me the term number, and I will give you its value.”

🔗 Recursive rule

Build the sequence step by step.

You use one or more previous terms to find the next term.

Think: “Give me where I am now, and I will tell you how to move to the next term.”

1. What is an explicit rule?

An explicit rule gives the value of the n^th term directly from its position. You do not have to calculate all the terms before it.

Example: Suppose a_n = 3n + 2. To find the 10th term, simply put n = 10: a₁₀ = 3(10) + 2 = 32.

Notice what happened: we did not need to find terms 1 through 9 first. We went straight to term 10. That is the main strength of an explicit rule.

2. What is a recursive rule?

A recursive rule tells you how to find a term from one or more earlier terms. It therefore needs an initial term (or initial terms) before the rule can be used.

Example:
a₁ = 5
a_n = a_n-1 + 3, for n ≥ 2

This says: start with 5, then add 3 each time.

First term: a₁ = 5
Second term: a₂ = a₁ + 3 = 5 + 3 = 8
Third term: a₃ = a₂ + 3 = 8 + 3 = 11
Fourth term: a₄ = a₃ + 3 = 11 + 3 = 14

So the sequence is:

5, 8, 11, 14, 17, ...

Explicit vs recursive: the easiest comparison

Explicit rule	Recursive rule
Uses the term number directly.	Uses previous term(s).
Can jump directly to a term.	Builds terms one after another.
Usually needs only the formula.	Needs the starting term(s) as well as the rule.
Useful for finding a far-away term.	Useful for describing how a sequence grows.
Example: a_n = 3n + 2	Example: a₁ = 5, a_n = a_n-1 + 3

One sequence, two rules

This is the part that often confuses students. The same sequence can have both an explicit rule and a recursive rule.

Sequence: 5, 8, 11, 14, 17, ...

Explicit

a_n = 5 + 3(n − 1)

a_n = 3n + 2

For example, a₂₀ = 3(20) + 2 = 62.

Recursive

a₁ = 5

a_n = a_n−1 + 3

Start at 5 and add 3 repeatedly.

Both descriptions produce exactly the same sequence. They simply describe it from different viewpoints.

Arithmetic sequences: the most common example

In an arithmetic sequence, the difference between consecutive terms is constant. That constant is called the common difference, usually written as d.

Arithmetic sequence: 12, 17, 22, 27, 32, ...
Common difference: d = 5

Recursive rule:

a₁ = 12, a_n = a_n−1 + 5

Explicit rule:

a_n = a₁ + (n − 1)d

a_n = 12 + 5(n − 1) = 5n + 7

Geometric sequences work in the same way

In a geometric sequence, each term is obtained by multiplying the previous term by the same number. That number is called the common ratio, usually written as r.

Geometric sequence: 3, 6, 12, 24, 48, ...
Common ratio: r = 2

Recursive rule:

a₁ = 3, a_n = 2a_n−1

Explicit rule:

a_n = 3(2)ⁿ⁻¹

To find the 8th term explicitly, substitute n = 8: a₈ = 3(2)⁷ = 384. With the recursive rule, you would keep multiplying by 2 until you reach the 8th term.

A simple real-life way to remember the difference

🏃 Explicit = GPS shortcut

Imagine asking a GPS, “Take me directly to house number 50.” You give the destination and go there directly.

Explicit rule → direct access to the term.

🪜 Recursive = climbing stairs

You start on one step and move to the next step using the same instruction again and again.

Recursive rule → one step at a time.

When should you use each rule?

If you want to...	A useful choice	Why?
Find the 100th term quickly	Explicit	Put n = 100 directly into the formula.
Generate the first few terms	Recursive	Start with the initial term and repeatedly apply the rule.
Describe a constant-addition pattern	Either	Arithmetic sequences have both standard forms.
Describe a pattern based on earlier terms	Recursive	The rule naturally shows how one term depends on earlier terms.

Fibonacci: a famous recursive sequence

The Fibonacci sequence is a good example of why recursive rules are useful. Starting with 1 and 1, each new term is the sum of the two previous terms:

1, 1, 2, 3, 5, 8, 13, 21, ...

Its recursive rule can be written as:

a₁ = 1, a₂ = 1, a_n = a_n−1 + a_n−2, for n ≥ 3

Here, two previous terms are needed to create the next term. This is an important point: a recursive rule does not have to use only one previous term.

How to identify the rule in an exam

If the formula contains n and directly gives a_n, it is usually an explicit rule.
If a_n is written using a_n−1, a_n−2, or other earlier terms, it is a recursive rule.
If a recursive rule is given, look for the initial term(s). Without a starting value, the sequence cannot be generated.
For an arithmetic sequence, remember: add d for the recursive rule and use a₁ + (n − 1)d for the explicit rule.
For a geometric sequence, remember: multiply by r for the recursive rule and use a₁rⁿ⁻¹ for the explicit rule.

The easiest memory trick

Explicit = Exact position → exact term.

Recursive = Previous term → next term.

If you remember just those two lines, the difference between the two rules becomes much easier. The formulas may look complicated at first, but their basic idea is simple.

Frequently asked questions

What does “explicit” mean in a sequence?

It means the rule gives the value of a term directly from its position. For example, a_n = 3n + 2 lets you find any term by substituting its term number.

What does “recursive” mean?

It means the rule uses one or more earlier terms to produce the next term. A recursive definition therefore includes the starting value or values.

Can one sequence have both rules?

Yes. For many familiar sequences, an explicit rule and a recursive rule describe the same terms.

Which one is better?

Neither is automatically better. They are useful for different purposes. Explicit rules are convenient for jumping to a particular term, while recursive rules are useful for showing how the sequence is generated.

Quick recap

Concept	Easy meaning
Sequence	An ordered list of numbers.
Explicit rule	Find a term directly from its position.
Recursive rule	Find a new term from previous term(s).
Initial term	The starting value needed by a recursive rule.
Arithmetic sequence	Add the same number each time.
Geometric sequence	Multiply by the same number each time.