Recursive Formula: Definition, Examples & vs Explicit

Definitionnoun phrase

What is a recursive formula?

A recursive formula defines a sequence by giving its first term and a rule that builds each next term from the term or terms before it. For example, a₁ = 5 and a_n = a_{n − 1} + 4 make 5, 9, 13, 17. To find a later term, you compute every term before it, one step at a time.

In Spanish: fórmula recursiva
Also called: recursive rule, recursive definition, recurrence relation, recursive sequence
Grade level: Algebra 1

Algebra 1Algebra 2CCSS HSF-BF.A.2CCSS HSF-IF.A.3SATACTACCUPLACERPSAT

Recursive Formula in a picture

In each row you need the earlier terms to make the next one: add 4, double, or add the previous two (the Fibonacci sequence).

Formal definition and notation

A recursive formula for a sequence consists of one or more initial terms, such as a₁, together with a recurrence relation that expresses a_n in terms of earlier terms, such as a_n = f(a_{n − 1}) for n ≥ 2. Arithmetic sequences satisfy a_n = a_{n − 1} + d and geometric sequences satisfy a_n = r · a_{n − 1}. The Fibonacci sequence uses two previous terms: a_n = a_{n − 1} + a_{n − 2} with a₁ = a₂ = 1.

Symbol	Read it as	Meaning
a_n	a sub n	the nth term
a_{n − 1}	a sub n minus 1	the term just before a_n
a₁ = 5	a sub 1 equals 5	the starting value (initial condition)
f(n) = f(n − 1) + 4	f of n equals f of n minus 1, plus 4	the same rule in function notation

Key facts

A recursive formula always has two parts: a starting term and a rule. Without the start, the sequence is not determined.
To get a₁₀ recursively you need a₉, which needs a₈, and so on: it is slow for far-away terms.
An explicit formula computes a_n directly from n; a recursive one computes it from earlier terms.
Some sequences are easiest to describe recursively, such as Fibonacci: 1, 1, 2, 3, 5, 8, 13.
Spreadsheets and computer loops work recursively: each cell or step uses the one before it.
Sequences are functions on the counting numbers, so f(n) = f(n − 1) + 4 is the same idea in function notation.

Common recursive formulas

a_n = a_{n − 1} + d or a_n = r · a_{n − 1}

a₁: the first term, which must be given
d: the common difference of an arithmetic sequence
r: the common ratio of a geometric sequence

The matching explicit formulas are a_n = a₁ + (n − 1)d and a_n = a₁r^{n − 1}, which jump straight to any term.

Try it: next term = 2 × previous + 1

Interactive

The machine applies the rule a_n = 2a_{n − 1} + 1. Starting from a₁ = 1, it has made 3, 7 and 15. Type the last output back in to make the next term; this is exactly how a recursive formula works.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Recursive Formula examples, step by step

Example 1: List the terms

Warm-up

Write the first four terms of a₁ = 7, a_n = a_{n − 1} − 3.

a₁ = 7.
a₂ = 7 − 3 = 4, a₃ = 4 − 3 = 1.
a₄ = 1 − 3 = −2.

Answer: The first four terms are 7, 4, 1, -2.

Example 2: Neither arithmetic nor geometric

Core

Find a₄ for a₁ = 2, a_n = 3a_{n − 1} + 1.

a₂ = 3(2) + 1 = 7.
a₃ = 3(7) + 1 = 22.
a₄ = 3(22) + 1 = 67.

Answer: The fourth term is 67.

Example 3: From recursive to explicit

Test-style

A sequence has a₁ = 10 and a_n = a_{n − 1} + 6. Write an explicit formula and find a₅₀.

Adding 6 each time means arithmetic with d = 6.
Explicit form: a_n = 10 + (n − 1) · 6 = 6n + 4.
a₅₀ = 6(50) + 4 = 304, with no need to list 49 earlier terms.

Answer: a_n = 6n + 4, so the 50th term is 304.

Common mistakes

✗ MistakeWriting only a_n = a_{n − 1} + 4.

✓ Insteada₁ = 5, a_n = a_{n − 1} + 4.

The rule alone fits 5, 9, 13 and also 2, 6, 10. The first term pins down which sequence you mean.

✗ MistakeFor a_n = 2a_{n − 1} with a₁ = 3: a₅ = 2 · 5 = 10.

✓ Instead3, 6, 12, 24, 48, so a₅ = 48.

n is the position, not the input to the rule. The rule uses the previous term, so build up one term at a time.

✗ MistakeUsing a_n = a_{n − 1} + 3 for 2, 6, 18, 54.

✓ Insteada₁ = 2, a_n = 3a_{n − 1}.

Check differences and ratios. The differences 4, 12, 36 are not constant, but the ratio is always 3.

Not to be confused with

Recursive Formulavs.Explicit formula

An explicit formula gives a_n directly from n: a_n = 4n + 1 gives a₁₀₀ = 401 at once. A recursive formula gives a_n from the previous term, so you must step through all the terms before it.

Recursive Formulavs.Common difference

The common difference is a number, such as 4. The recursive formula is a rule that uses it: a_n = a_{n − 1} + 4.

Quick check: 3 questions

Question 1 of 3

Which recursive formula gives 4, 12, 36, 108, …?

Show answer and explanation

Answer: B. Each term is 3 times the one before, starting at 4. Adding 8 works only for the first step, the third choice swaps start and ratio, and the last is an explicit formula (and gives 12 for n = 1).

Question 2 of 3

If a₁ = 2 and a_n = (a_{n − 1})², what is a₃?

Show answer and explanation

Answer: C. a₂ = 2² = 4 and a₃ = 4² = 16. 4 stops at a₂, 8 is 2³, and 9 squares the position 3.

Question 3 of 3

The sequence 1, 1, 2, 3, 5, 8 continues by the rule a_n = a_{n − 1} + a_{n − 2}. What is the next term?

Show answer and explanation

Answer: D. Add the two previous terms: 5 + 8 = 13. 16 doubles 8, 11 adds 3 to 8, and 10 adds 2.

Recursive Formula: frequently asked questions

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

A recursive formula tells you how to get from one term to the next, such as a_n = a_{n − 1} + 5 with a₁ = 2. An explicit formula tells you any term from its position, such as a_n = 5n − 3. Both describe 2, 7, 12, 17, but only the explicit formula finds a₂₀₀ in one step.

How do you write a recursive formula for a sequence?

First, give the first term. Then find how each term relates to the one before it: a constant difference means a_n = a_{n − 1} + d, and a constant ratio means a_n = r · a_{n − 1}. For 50, 45, 40, the recursive formula is a₁ = 50, a_n = a_{n − 1} − 5.

Why do recursive formulas need a starting value?

The rule only says how to move from one term to the next; it does not say where to begin. The rule “add 3” could produce 1, 4, 7 or 10, 13, 16. Giving a₁ fixes the first term, and then every later term follows. Rules that use two earlier terms, like Fibonacci, need two starting values.

Practice and tools

Sequence CalculatorFind terms and partial sums of arithmetic and geometric sequences Fibonacci CalculatorGenerate the most famous recursive sequence

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.A.2, CCSS HSF-IF.A.3

Cite this entry

Effortless Math Team. “Recursive Formula.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/recursive-formula/

Flashcard

TermRecursive Formulaa_n = a_{n − 1} + d

DefinitionA first term plus a rule that makes each term from the one before. a₁ = 5, a_n = a_{n − 1} + 4 gives 5, 9, 13, 17.

Study the full deck

Browse all terms in the Math Dictionary · More functions terms · Random term

SequenceA sequence is an ordered list of numbers that follows a rule, such as 3, 7, 11, 15, …. Each number is a term, and its position is 1st, 2nd, 3rd and so on. You can describe a sequence with a formula for the nth term, a_n, or by how each term comes from the one before it.Functions · Grade 8, Algebra 1, Algebra 2RelationA relation is any set of ordered pairs that links inputs to outputs, such as {(1, 4), (2, 5), (2, 7)}. It can be shown as a list, table, mapping diagram or graph. A relation in which every input has exactly one output is a function; this one is not, because 2 has two outputs.Functions · Grade 8, Algebra 1Arithmetic SequenceAn arithmetic sequence is a list of numbers in which you add the same amount each time to get the next term. That fixed amount is the common difference, d. In 4, 7, 10, 13, … the common difference is 3. The nth term is a_n = a₁ + (n − 1)d.Functions · Grade 8, Algebra 1, Algebra 2Geometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is a_n = a₁ · r^{n − 1}. The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2, PrecalculusFunction NotationFunction notation is the way of writing a function as f(x), read “f of x”: f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Functions · Grade 8, Algebra 1, Algebra 2Counting NumbersCounting numbers are the numbers you say when you count objects one by one: 1, 2, 3, 4, 5 and so on, without end. They start at 1, not 0, and include no fractions, decimals or negative numbers. Adding 0 to the counting numbers gives the whole numbers.Number System · Kindergarten, Grade 1, Grade 2FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Common DifferenceThe common difference is the fixed amount added to each term of an arithmetic sequence to get the next term, written d. Find it by subtracting any term from the term after it: in 7, 11, 15, 19, d = 11 − 7 = 4. A positive d makes the terms increase, a negative d makes them decrease.Functions · Grade 8, Algebra 1, Algebra 2DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SeriesA series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written S_n. An infinite series has a sum only if its partial sums approach a fixed number.Functions · Algebra 2, Precalculus, CalculusSigma NotationSigma notation is a compact way to write a long sum using the Greek capital letter Σ (sigma). In Σ⁴_k=1 k², the index k starts at the number below sigma, stops at the number above it, and each value is put into the rule k²: 1 + 4 + 9 + 16 = 30.Functions · Algebra 2, Precalculus, CalculusExponential GrowthExponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)^t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.Functions · Algebra 1, Algebra 2Common RatioThe common ratio is the fixed number you multiply each term of a geometric sequence by to get the next term. Find it by dividing any term by the term before it: r = a_na_{n − 1}. In 5, 20, 80, 320, the common ratio is 4. It can be a fraction or negative, but never 0.Functions · Algebra 1, Algebra 2

Recursive Formula

What is a recursive formula?

Recursive Formula in a picture

Formal definition and notation

Key facts

Common recursive formulas

Try it: next term = 2 × previous + 1

Recursive Formula examples, step by step

Example 1: List the terms

Example 2: Neither arithmetic nor geometric

Example 3: From recursive to explicit

Common mistakes

Not to be confused with

Quick check: 3 questions

Recursive Formula: frequently asked questions

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

How do you write a recursive formula for a sequence?

Why do recursive formulas need a starting value?

Related terms

Practice and tools

Sources

Flashcard

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