What is the difference between a recursive and an explicit formula?

A recursive formula requires you to know the previous term to find the next one (e.g., aₙ = aₙ₋₁ + 3). An explicit formula allows you to calculate any term in the sequence directly using its position, n (e.g., aₙ = 3n + 2).

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Recursive Formula

Sequences of numbers are fundamental building blocks in algebra and calculus. A recursive formula defines each term in relation to the ones that came before. For instant progression generation, try our arithmetic sequence calculator or the geometric sequence calculator.

6 min read

Blake BoegeFounder, Calculator AnswersPublished May 29, 2026

What is a recursive formula?

A recursive formula uses the term immediately before (and sometimes older terms) to calculate the next term in a sequence. It acts like a step-by-step instruction set where you start at a base value and apply a repeat operation over and over.

Recursive formulas have two mandatory parts:

The initial condition: The value of the first term (a₁ = value).
The recurrence relation: The formula to find a_n using the previous term a_n−1.

Common Sequence Recurrence Rules

Sequence Type	Recurrence Rule	Variables Explained
Arithmetic	`a`_n = `a`_n−1 + `d`	`a`_n−1 = previous term, `d` = common difference
Geometric	`a`_n = `r` × `a`_n−1	`a`_n−1 = previous term, `r` = common ratio
Fibonacci	`a`_n = `a`_n−1 + `a`_n−2	`a`₁ = 1, `a`₂ = 1, sums the two prior terms

Worked Examples

Example 1: Arithmetic Progression

Find the first four terms of the sequence defined recursively by:

a₁ = 5

a_n = a_n−1 + 3

Term 1: a₁ = 5
Term 2: a₂ = a₁ + 3 = 5 + 3 = 8
Term 3: a₃ = a₂ + 3 = 8 + 3 = 11
Term 4: a₄ = a₃ + 3 = 11 + 3 = 14
Sequence: 5, 8, 11, 14...

Example 2: Geometric Progression

Find the first four terms of the sequence defined recursively by:

a₁ = 2

a_n = 3 × a_n−1

Term 1: a₁ = 2
Term 2: a₂ = 3 × a₁ = 3 × 2 = 6
Term 3: a₃ = 3 × a₂ = 3 × 6 = 18
Term 4: a₄ = 3 × a₃ = 3 × 18 = 54
Sequence: 2, 6, 18, 54...

Common Mistakes with Recursive Formulas

Subtracting from the term value instead of the index: The expression a_n−1 refers to the term at position n − 1. It does not mean you take the value of a_n and subtract 1 from it.
Calculating high-index terms manually: Trying to calculate the 100th term (a₁₀₀) using a recursive formula is tedious and prone to arithmetic mistakes because you must compute the first 99 terms first. In these cases, convert the rule to an explicit formula instead.
Omitting the starting term (a₁): A recursive formula without a starting term is incomplete and describes infinitely many different sequences.

Run the numbers

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Frequently asked questions

A recursive formula is a rule that defines each term of a sequence based on one or more preceding terms. To use a recursive formula, you must always be given a starting value, typically the first term (a₁).

A recursive formula requires you to know the previous term to find the next one (e.g., a_n = a_n−1 + 3). An explicit formula allows you to calculate any term in the sequence directly using its position, n (e.g., a_n = 3n + 2).

Recursive formulas are highly intuitive for describing natural growth processes, computer programming loops, and sequences like the Fibonacci numbers, where each number is the sum of the two preceding ones.

Without a starting term (or base case), a recursive formula cannot define a sequence. The recursive rule only tells you how to step from one term to the next; it does not tell you where to begin.

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