Common Difference in Math: Definition, Formula & Examples

Definitionnoun phrase

What is the common difference in a sequence?

The 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.

In Spanish: diferencia común
Also called: d, constant difference, common difference of an arithmetic sequence
Grade level: Algebra 1

Grade 8Algebra 1Algebra 2CCSS HSF-BF.A.2CCSS HSF-LE.A.1.aCCSS HSF-LE.A.2SATACTASVABGEDACCUPLACER

Common Difference in a picture

The common difference is the same jump every time: +4 in the first sequence, −3 in the second.

Formal definition and notation

For an arithmetic sequence a₁, a₂, a₃, …, the common difference is the constant d = a_{n + 1} − a_n, the same for every n ≥ 1. The terms then satisfy a_n = a₁ + (n − 1)d, and for any positions m ≠ k, d = a_m − a_km − k. A sequence has a common difference only if it is arithmetic.

Symbol	Read it as	Meaning
d	d	the common difference
d = a_{n + 1} − a_n	d equals a sub n plus 1 minus a sub n	next term minus current term
a_n = a₁ + (n − 1)d	a sub n equals a sub 1 plus n minus 1 times d	the explicit formula that uses d

Key facts

Always later minus earlier. In 20, 17, 14, d = 17 − 20 = −3, not 3.
d can be negative, a fraction or a decimal: 1, 1.5, 2, 2.5 has d = 0.5.
If d = 0, every term is the same: 6, 6, 6, … is arithmetic.
Check every pair. If the differences change (2, 4, 8, 16), there is no common difference; the sequence is not arithmetic.
Plotted as points (n, a_n), an arithmetic sequence lies on a line, and d is its slope.

Finding the common difference

d = a_{n + 1} − a_n = a_m − a_km − k

a_{n + 1} − a_n: any term minus the term just before it
a_m, a_k: any two terms, in positions m and k
m − k: how many steps apart the two terms are

The second form works when the terms are not next to each other: the total change divided by the number of steps.

Try it: add the common difference

Interactive

This machine adds the common difference 4. Type the last output back in and press Run to build 7, 11, 15, 19 … one term at a time.

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

Common Difference examples, step by step

Example 1: Find d

Warm-up

Find the common difference of 4, 9, 14, 19, …

Subtract each term from the next: 9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5.
All the differences are 5, so the sequence is arithmetic with d = 5.

Answer: The common difference is 5.

Example 2: Negative and decimal

Core

Find d and the 10th term of 3, 2.5, 2, 1.5, …

d = 2.5 − 3 = −0.5. Check: 2 − 2.5 = −0.5.
Use a_n = a₁ + (n − 1)d with n = 10.
a₁₀ = 3 + 9(−0.5) = 3 − 4.5.
a₁₀ = −1.5.

Answer: d = −0.5 and the 10th term is −1.5.

Example 3: Terms that are not next to each other

Test-style

In an arithmetic sequence, a₃ = 11 and a₈ = 36. What is the common difference, and what is a₁?

From term 3 to term 8 is 8 − 3 = 5 steps.
Total change: 36 − 11 = 25.
d = 255 = 5.
Step back two terms from a₃: a₁ = 11 − 2 × 5 = 1.

Answer: d = 5 and a₁ = 1.

Common mistakes

✗ MistakeFor 20, 17, 14, d = 20 − 17 = 3.

✓ Insteadd = 17 − 20 = −3.

Subtract in the direction the sequence goes: later term minus earlier term. A decreasing sequence has a negative common difference.

✗ Mistakea₃ = 11 and a₈ = 36, so d = 36 − 11 = 25.

✓ Insteadd = 36 − 118 − 3 = 5.

The 25 is spread over 5 steps. Divide the change by the number of steps between the positions.

✗ Mistake2, 4, 8, 16 has common difference 2.

✓ InsteadThe differences are 2, 4 and 8, so there is no common difference. (It is geometric, with ratio 2.)

One pair is not enough. A common difference must be the same for every pair of neighbors.

Not to be confused with

Common Differencevs.Common ratio

The common ratio is what you multiply by in a geometric sequence (3, 6, 12 has ratio 2). The common difference is what you add (3, 5, 7 has difference 2).

Common Differencevs.Arithmetic sequence

The arithmetic sequence is the whole list; the common difference is the one number that builds it. 5, 9, 13, 17 is an arithmetic sequence with d = 4.

Common Differencevs.Geometric sequence

A geometric sequence has no common difference; its terms change by a common ratio instead.

Quick check: 3 questions

Question 1 of 3

What is the common difference of 15, 11, 7, 3, …?

Show answer and explanation

Answer: B. 11 − 15 = −4, and every later pair also gives −4. 4 subtracts in the wrong order, −3 confuses a term with the difference, and 1115 is a ratio.

Question 2 of 3

An arithmetic sequence has a₁ = 6 and d = 7. What is a₂₀?

Show answer and explanation

Answer: C. a₂₀ = 6 + 19 × 7 = 6 + 133 = 139. 146 uses 20 steps instead of 19, 140 is 20 × 7, and 133 forgets to add the first term.

Question 3 of 3

Which sequence has a common difference?

Show answer and explanation

Answer: C. 12, 1, 32, 2 adds 12 every time. 1, 3, 9, 27 multiplies by 3, and the other two have differences that change (1, 2, 3 and 1, 2, 3).

Common Difference: frequently asked questions

How do you find the common difference?

Subtract any term from the term that comes right after it: d = a₂ − a₁. Then check a second pair to make sure the difference really is constant. For 12, 19, 26, 33 you get 19 − 12 = 7 and 26 − 19 = 7, so d = 7. If two terms are far apart, divide their difference by the number of steps between them.

Can the common difference be negative?

Yes. A negative common difference means the sequence decreases. In 50, 44, 38, 32, each term is 6 less than the one before it, so d = −6. The common difference can also be a fraction or a decimal, and it is 0 when every term is the same.

Is the common difference the same as slope?

They match. If you plot an arithmetic sequence as points (n, a_n), the points lie on a straight line, and each step of 1 in n raises the value by d. So the common difference is the slope of that line, and the explicit formula a_n = a₁ + (n − 1)d is a linear function of n.

Practice and tools

Sequence CalculatorFind d, any term, or the sum of an arithmetic sequence

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.A.2, CCSS HSF-LE.A.1.a, CCSS HSF-LE.A.2

Cite this entry

Effortless Math Team. “Common Difference.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/common-difference/

Flashcard

TermCommon Differenced = a_{n + 1} − a_n

DefinitionThe fixed amount added each step in an arithmetic sequence. 7, 11, 15, 19 has d = 4; 20, 17, 14 has d = −3.

Study the full deck

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

Arithmetic 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 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 3SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1NumberA 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 2RatioA 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 8Common 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 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, PrecalculusLinear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8SequenceA 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 2Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1FunctionA 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 2

Common Difference

What is the common difference in a sequence?

Common Difference in a picture

Formal definition and notation

Key facts

Finding the common difference

Try it: add the common difference

Common Difference examples, step by step

Example 1: Find d

Example 2: Negative and decimal

Example 3: Terms that are not next to each other

Common mistakes

Not to be confused with

Quick check: 3 questions

Common Difference: frequently asked questions

How do you find the common difference?

Can the common difference be negative?

Is the common difference the same as slope?

Related terms

Practice and tools

Sources

Flashcard

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