Arithmetic Sequence in Math: Definition, Formula & Examples

Definitionnoun phrase

What is an arithmetic sequence in math?

An 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.

In Spanish: sucesión aritmética
Also called: arithmetic progression, AP, linear sequence
Grade level: Algebra 1

Grade 8Algebra 1Algebra 2CCSS HSF-BF.A.2CCSS HSF-IF.A.3SATACTASVABGEDHiSETACCUPLACER

Arithmetic Sequence in a picture

Each term is 3 more than the one before, so d = 3. Because the same amount is added each time, the points (n, a_n) fall on a straight line.

Formal definition and notation

A sequence a₁, a₂, a₃, … is arithmetic when a_n − a_n−1 = d for every n ≥ 2, for some constant d. Recursively, a_n = a_n−1 + d; explicitly, a_n = a₁ + (n − 1)d. As a function of the term number n, an arithmetic sequence is linear with slope d, defined only on the positive integers.

Symbol	Read it as	Meaning
a_n	a sub n	the nth term
a₁	a sub 1	the first term
d	d	the common difference (it can be negative or a fraction)
S_n	S sub n	the sum of the first n terms (an arithmetic series)

Key facts

Test: subtract each term from the next one. If every difference is the same, the sequence is arithmetic.
A positive d makes the terms grow, a negative d makes them shrink (20, 14, 8, 2, … has d = −6), and d = 0 gives a constant sequence.
Graphed as points (n, a_n), the terms lie on a straight line with slope d, so arithmetic sequences grow linearly.
The explicit formula can be simplified to slope-intercept style: a_n = 4 + (n − 1)3 = 3n + 1.
Compare: an arithmetic sequence adds the same number each time; a geometric sequence multiplies by the same number each time.

Arithmetic sequence formulas

a_n = a₁ + (n − 1)d S_n = n(a₁ + a_n)2

a_n: the term in position n
a₁: the first term
d: the common difference, a₂ − a₁
n: the term number (1, 2, 3, …)

You add d only n − 1 times to get from the first term to the nth term, which is why the formula has n − 1, not n.

Try it: term number in, term out

Interactive

This machine runs the rule for the sequence 3, 7, 11, 15, …: multiply the term number by 4, then subtract 1. Feed it term numbers and watch the outputs go up by 4 each time.

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

Arithmetic Sequence examples, step by step

Example 1: Find the next terms

Warm-up

Find the common difference and the next two terms of 15, 11, 7, 3, …

Subtract consecutive terms: 11 − 15 = −4, 7 − 11 = −4, 3 − 7 = −4. So d = −4.
Next term: 3 + (−4) = −1.
Term after that: −1 + (−4) = −5.

Answer: d = −4, and the next two terms are −1 and −5.

Example 2: Find a far-away term

Core

Find the 40th term of the arithmetic sequence 6, 11, 16, 21, …

First term a₁ = 6, common difference d = 11 − 6 = 5.
Use a_n = a₁ + (n − 1)d with n = 40: a₄₀ = 6 + 39 × 5.
39 × 5 = 195, so a₄₀ = 6 + 195 = 201.

Answer: The 40th term is 201.

Example 3: Which term is it, and what is the total?

Test-style

A theater has 22 seats in the first row, and each row after that has 3 more seats than the row in front of it. Which row has 79 seats, and how many seats are in all the rows up to and including it?

The rows form an arithmetic sequence with a₁ = 22 and d = 3.
Set 22 + (n − 1) × 3 = 79. Then (n − 1) × 3 = 57, so n − 1 = 19 and n = 20.
Sum of 20 rows: S₂₀ = 20(22 + 79)2.
S₂₀ = 10 × 101 = 1010.

Answer: Row 20 has 79 seats, and rows 1 to 20 hold 1,010 seats in all.

Common mistakes

✗ Mistake40th term of 6, 11, 16, …: 6 + 40 × 5 = 206.

✓ Insteada₄₀ = 6 + 39 × 5 = 201.

From the 1st term to the 40th term there are only 39 steps, so you add d only n − 1 times.

✗ MistakeFor 15, 11, 7, 3, the common difference is 4.

✓ Insteadd = 11 − 15 = −4.

Always subtract a term minus the one before it. A decreasing sequence has a negative common difference.

✗ Mistake2, 4, 8, 16, … is arithmetic because it goes up in a pattern.

✓ InsteadThe differences are 2, 4, 8, which are not equal, so it is not arithmetic. It multiplies by 2 each time (geometric).

Arithmetic means the same number is added each time. A different kind of pattern does not count.

Not to be confused with

Arithmetic Sequencevs.Geometric sequence

An arithmetic sequence adds the same number each time (5, 8, 11, 14); a geometric sequence multiplies by the same number each time (5, 10, 20, 40).

Arithmetic Sequencevs.Common difference

The arithmetic sequence is the whole list of numbers; the common difference d is the single number added each step. In 5, 8, 11, … the sequence goes on forever, and d = 3.

Arithmetic Sequencevs.Arithmetic series

A sequence is a list (2, 5, 8, 11); a series is the sum of its terms (2 + 5 + 8 + 11 = 26).

Quick check: 3 questions

Question 1 of 3

Which list is an arithmetic sequence?

Show answer and explanation

Answer: C. 10, 7, 4, 1 subtracts 3 every time, so d = −3. The squares 1, 4, 9, 16 have differences 3, 5, 7; 3, 6, 12, 24 doubles (geometric); and 1, 1, 2, 3 is the Fibonacci pattern.

Question 2 of 3

What is the 25th term of 8, 14, 20, 26, …?

Show answer and explanation

Answer: A. a₂₅ = 8 + 24 × 6 = 8 + 144 = 152. 158 uses 25 × 6 (one step too many), 150 is 25 × 6 alone, and 200 is 25 × 8.

Question 3 of 3

Which explicit formula gives the sequence 9, 5, 1, −3, …?

Show answer and explanation

Answer: D. a₁ = 9 and d = −4, so a_n = 9 + (n − 1)(−4) = 13 − 4n. Check: 13 − 4 = 9, 13 − 8 = 5. The formula 9 − 4n gives 5 as the first term, because it adds d one time too many.

Arithmetic Sequence: frequently asked questions

How do you know if a sequence is arithmetic?

Subtract each term from the term after it. If all the differences are the same number, the sequence is arithmetic and that number is the common difference. For 7, 12, 17, 22 every difference is 5, so it is arithmetic. For 1, 3, 7, 15 the differences are 2, 4, 8, so it is not.

What is the difference between the recursive and explicit formulas?

The recursive formula tells you how to get each term from the one before: a₁ = 4 and a_n = a_n−1 + 3. The explicit formula jumps straight to any term from its position: a_n = 4 + (n − 1) × 3. Use the explicit formula when you need a term far down the list, such as the 100th.

How do you find the sum of an arithmetic sequence?

Add the first and last terms, multiply by the number of terms, and divide by 2: S_n = n(a₁ + a_n)2. For 1 + 2 + 3 + … + 100, that is 100 × 1012 = 5050. It works because pairing the first and last terms, the second and second-to-last, and so on, gives equal sums.

Practice and tools

Sequence CalculatorFind any term and the formula of an arithmetic sequence Summation CalculatorAdd up the terms of an arithmetic series

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. “Arithmetic Sequence.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/arithmetic-sequence/

Flashcard

TermArithmetic Sequencea_n = a₁ + (n − 1)d

DefinitionA list where the same number is added each time. 4, 7, 10, 13 has common difference 3.

Study the full deck

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Arithmetic Sequence

What is an arithmetic sequence in math?

Arithmetic Sequence in a picture

Formal definition and notation

Key facts

Arithmetic sequence formulas

Try it: term number in, term out

Arithmetic Sequence examples, step by step

Example 1: Find the next terms

Example 2: Find a far-away term

Example 3: Which term is it, and what is the total?

Common mistakes

Not to be confused with

Quick check: 3 questions

Arithmetic Sequence: frequently asked questions

How do you know if a sequence is arithmetic?

What is the difference between the recursive and explicit formulas?

How do you find the sum of an arithmetic sequence?

Related terms

Practice and tools

Sources

Flashcard

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