What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers that changes by the same amount each time. That repeated change is called the common difference. For example, \(5, 8, 11, 14, \dots\) is arithmetic because each term is \(3\) more than the term before it.
Arithmetic sequences are useful whenever a pattern grows or decreases by equal steps. They appear in row seating, repeated savings amounts, numbered patterns, simple schedules, and many algebra problems. Once you know the first term and the common difference, you can find a later term without writing every earlier term.
An arithmetic series is closely related, but it answers a different question. A sequence lists the terms. A series adds terms together. For example, the sequence \(5, 8, 11\) has the series sum \(5 + 8 + 11 = 24\).
Why Arithmetic Sequences Matter
Arithmetic sequences help turn a repeated pattern into a formula. Instead of counting forward one step at a time, you can calculate the term you need directly.
This matters because arithmetic patterns are often simple but can become tedious when the term number is large. Finding the \(8\)th term by listing terms is manageable. Finding the \(800\)th term is much easier with a formula.
Arithmetic series are useful for totals. If a pattern gives the number of items in each row, the points earned each round, or the amount added during each step, the series formula gives the total across a chosen number of terms.
Key Terms to Know
-
Term: One value in the sequence. In \(5, 8, 11, 14, \dots\), each number is a term.
-
First term: The starting value, written as \(a_1\).
-
Term number: The position of a term in the sequence. The first term has term number \(1\), the second has term number \(2\), and so on.
-
Common difference: The fixed amount added to move from one term to the next, written as \(d\).
-
nth term: The term at position \(n\), written as \(a_n\).
-
Arithmetic series: The sum of terms from an arithmetic sequence.
-
Finite sequence: A sequence with a limited number of terms.
-
Sigma notation: A compact way to write a sum, using the symbol \(\sum\).
How Arithmetic Sequences Work
An arithmetic sequence starts with \(a_1\) and repeatedly adds the same common difference \(d\). That means the second term is \(a_1 + d\), the third term is \(a_1 + 2d\), and the fourth term is \(a_1 + 3d\).
The important idea is that the \(n\)th term is not \(n\) steps away from the first term. It is \(n - 1\) steps away, because the first term is already step zero.
The nth-term formula is:
$$
a_n = a_1 + (n - 1)d
$$
Where:
-
\(a_n\) = the term at position \(n\)
-
\(a_1\) = the first term
-
\(n\) = the term number
-
\(d\) = the common difference
The common difference can be found by subtracting one term from the next:
$$
d = a_k - a_{k - 1}
$$
For example, in \(12, 17, 22, 27, \dots\), the common difference is:
$$
d = 17 - 12 = 5
$$
Because the difference is positive, the sequence increases. If \(d\) is negative, the sequence decreases. If \(d = 0\), every term is the same.
How Arithmetic Series Sums Work
An arithmetic series adds terms from an arithmetic sequence. If you add the first \(m\) terms, the result can be written as \(S_m\):
$$
S_m = a_1 + a_2 + a_3 + \cdots + a_m
$$
One way to understand the arithmetic series formula is to pair the first and last terms, the second and next-to-last terms, and so on. Each pair has the same total. That is why the sum can be found by multiplying the number of terms by the average of the first and last term:
$$
S_m = \frac{m}{2}(a_1 + a_m)
$$
If you know the first term and common difference, you can replace \(a_m\) with \(a_1 + (m - 1)d\). This gives the common sum formula:
$$
S_m = \frac{m}{2}\left[2a_1 + (m - 1)d\right]
$$
Where:
-
\(S_m\) = the sum of the first \(m\) terms
-
\(m\) = the number of terms included in the sum
-
\(a_1\) = the first term
-
\(d\) = the common difference
The letter does not matter as much as the meaning. Textbooks often use \(n\) for both term formulas and sum formulas, but this calculator lets you enter a term number and a separate number of terms for the sum. In this article, \(n\) refers to the term you want to find, while \(m\) refers to how many terms you want to add.
Examples of Arithmetic Sequences in Practice
Example 1: Finding an nth Term
Suppose the first term is \(5\), the common difference is \(3\), and you want the \(8\)th term.
Use the nth-term formula:
$$
a_n = a_1 + (n - 1)d
$$
Substitute \(a_1 = 5\), \(d = 3\), and \(n = 8\):
$$
a_8 = 5 + (8 - 1)3
$$
$$
a_8 = 5 + 7 \times 3
$$
$$
a_8 = 26
$$
The \(8\)th term is \(26\). The first six terms are \(5, 8, 11, 14, 17, 20\).
Example 2: Finding a Finite Sum
Use the same sequence with \(a_1 = 5\) and \(d = 3\). To find the sum of the first \(8\) terms, use:
$$
S_m = \frac{m}{2}\left[2a_1 + (m - 1)d\right]
$$
Substitute \(m = 8\):
$$
S_8 = \frac{8}{2}\left[2(5) + (8 - 1)3\right]
$$
$$
S_8 = 4(10 + 21)
$$
$$
S_8 = 124
$$
The sum of the first \(8\) terms is \(124\).
Example 3: A Decreasing Sequence
A negative common difference creates a decreasing arithmetic sequence. Suppose \(a_1 = 20\), \(d = -4\), and \(n = 6\).
$$
a_6 = 20 + (6 - 1)(-4)
$$
$$
a_6 = 20 - 20
$$
$$
a_6 = 0
$$
The first six terms are \(20, 16, 12, 8, 4, 0\). The sum of those six terms is:
$$
S_6 = \frac{6}{2}\left[2(20) + (6 - 1)(-4)\right]
$$
$$
S_6 = 3(40 - 20)
$$
$$
S_6 = 60
$$
Example 4: A Constant Sequence
If the common difference is \(0\), every term is the same. Suppose \(a_1 = 7\), \(d = 0\), and you want the \(10\)th term.
$$
a_{10} = 7 + (10 - 1)(0)
$$
$$
a_{10} = 7
$$
The sum of the first \(5\) terms is:
$$
S_5 = \frac{5}{2}\left[2(7) + (5 - 1)(0)\right]
$$
$$
S_5 = \frac{5}{2}(14)
$$
$$
S_5 = 35
$$
Example 5: A Real-World Row Pattern
Imagine an auditorium where the first row has \(24\) seats and each next row has \(4\) more seats than the row before it. The number of seats per row forms an arithmetic sequence with \(a_1 = 24\) and \(d = 4\).
The \(12\)th row has:
$$
a_{12} = 24 + (12 - 1)4
$$
$$
a_{12} = 24 + 44
$$
$$
a_{12} = 68
$$
The total number of seats in the first \(12\) rows is:
$$
S_{12} = \frac{12}{2}(24 + 68)
$$
$$
S_{12} = 6 \times 92
$$
$$
S_{12} = 552
$$
So the first \(12\) rows contain \(552\) seats in total.
How to Interpret the Result
The \(a_n\) result is the value of the term at the term number you entered. If you enter \(n = 8\), the result is the \(8\)th term, not the sum of eight terms.
The sum result is the total of the first number of terms requested for the sum. This can be different from the term number \(n\). For example, you may calculate the \(20\)th term while summing only the first \(10\) terms. In that case, \(a_{20}\) and \(S_{10}\) answer different questions.
The first-six-terms preview shows the beginning of the pattern. It helps you check whether the sequence is increasing, decreasing, or constant, but it is only a preview. It does not show every term up to a large term number.
The trend label follows the sign of the common difference:
-
If \(d > 0\), the sequence is increasing.
-
If \(d < 0\), the sequence is decreasing.
-
If \(d = 0\), the sequence is constant.
The calculator preserves accepted decimal inputs and term and sum results exactly in decimal-place arithmetic. Very large or very small exact values may be shown in scientific notation. The graph is a visual preview, so points whose differences are smaller than browser drawing precision can overlap even when the displayed terms remain distinct.
Common Mistakes and Misconceptions
A common mistake is using \(n\) instead of \(n - 1\) in the nth-term formula. The first term is already at position \(1\), so the number of jumps from \(a_1\) to \(a_n\) is \(n - 1\).
Another mistake is confusing an arithmetic sequence with a geometric sequence. An arithmetic sequence adds the same difference each time. A geometric sequence multiplies by the same ratio each time. The sequence \(5, 10, 20, 40, \dots\) is geometric, not arithmetic, because it multiplies by \(2\).
It is also easy to confuse the term number with the number of terms in a sum. The term number tells you which term to find. The number of terms for the sum tells you how many starting terms to add.
Do not assume the graph preview shows the entire sequence. It shows the first six terms only. A sequence can look simple in the preview and still produce a very large value at a high term number.
Finally, copied results preserve the calculator's displayed exact decimal value. Keep the original input values and formulas visible when you need to document or independently check the work.
When to Use Arithmetic Sequences
Use arithmetic sequences when a pattern changes by equal additions or subtractions. They are useful for:
-
Finding a term far ahead in a repeated pattern.
-
Checking whether a list of values has a constant difference.
-
Modeling row counts, step counts, or evenly changing quantities.
-
Finding the total of a finite number of terms.
-
Comparing recursive thinking, where each term depends on the previous term, with explicit formulas, where a term can be found directly.
If the change is not constant, an arithmetic sequence may not be the right model. For example, compound growth, repeated doubling, and percentage growth are usually geometric or exponential patterns instead.
Limitations and Things to Keep in Mind
Arithmetic sequence formulas assume the common difference is constant. If the pattern changes by different amounts, the formulas may give misleading results.
The calculator works with unitless numeric values. If your real-world problem uses seats, points, dollars, meters, or another unit, interpret the final number in the context of your problem.
The term number and the number of terms for the sum must be positive whole numbers. Decimal, negative, zero, and scientific-notation entries are not valid for these two fields.
The first term and common difference can be decimal numbers, and scientific notation is accepted for those fields. However, symbolic fractions, algebraic expressions, text, and units are not accepted as numeric inputs.
To keep exact-decimal calculations responsive, each first-term and common-difference entry is limited to 10,000 characters, 100 significant digits after insignificant leading and trailing zeroes are trimmed, and an absolute exponent and effective decimal-scale limit of 1,000. These limits apply before the calculator creates its exact decimal representation.
The supported input range is limited. The first term and common difference must have absolute value no greater than \(1{,}000{,}000{,}000{,}000\). The term number and the number of terms for the sum must each be from \(1\) through \(1{,}000{,}000\).
The calculator does not solve for missing variables. It calculates the requested term and finite sum from the first term, common difference, term number, and sum count that you provide.
The graph and first-terms preview are limited to the first six terms. They are meant for quick visual checking, not for exporting a full sequence.
The calculator preserves accepted decimal inputs and results with decimal-place arithmetic. A graph point may visually overlap another point at an extreme relative scale, so use the displayed first-terms preview, formula substitution, or copied result to compare those values exactly.
How to Use This Calculator
-
Enter the first term of the arithmetic sequence.
-
Enter the common difference between consecutive terms.
-
Enter the term number \(n\) for the specific term you want to find.
-
Enter the number of terms to include in the sum result.
-
Review the nth-term result, finite-sum result, first-six-terms preview, trend label, graph, and formula substitutions.
-
Use the example buttons to test sample values, the clear button to reset the fields, or the graph download option to save the preview image.
Frequently Asked Questions
What is the formula for an arithmetic sequence?
The nth-term formula is \(a_n = a_1 + (n - 1)d\). It uses the first term, the common difference, and the term number to find a specific term without listing every earlier term.
Why does the formula use \(n - 1\)?
The first term is already at position \(1\), so there are zero jumps from \(a_1\) to itself. To reach the second term there is one jump, to reach the third term there are two jumps, and to reach the \(n\)th term there are \(n - 1\) jumps.
What is the difference between an arithmetic sequence and an arithmetic series?
An arithmetic sequence is the list of terms. An arithmetic series is the sum of terms from that list. For example, \(4, 7, 10\) is a sequence, while \(4 + 7 + 10 = 21\) is a series sum.
Can the common difference be negative or zero?
Yes. A negative common difference creates a decreasing sequence, while a common difference of zero creates a constant sequence where every term is the same.
Why are the term number and sum count separate?
They answer different questions. The term number finds one selected term, such as \(a_{20}\). The sum count finds the total of the first chosen number of terms, such as \(S_{10}\).
Does the graph show every term up to \(n\)?
No. The graph is a preview of the first six terms. It helps you see the early pattern, but it does not display every term for large values of \(n\) or for the sum count.
Sources and References
Books
-
Jay Abramson. College Algebra 2e. OpenStax, 2021. Chapter 9, Section 9.2, “Arithmetic Sequences.” https://openstax.org/books/college-algebra-2e/pages/9-2-arithmetic-sequences
-
Lynn Marecek and Andrea Honeycutt Mathis. Intermediate Algebra 2e. OpenStax, 2020. Chapter 12, Section 12.2, “Arithmetic Sequences.” https://openstax.org/books/intermediate-algebra-2e/pages/12-2-arithmetic-sequences
Online and Open Educational Sources
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Mathematics LibreTexts. “9.2: Arithmetic Sequences and Series.” Updated October 6, 2021. Accessed June 27, 2026. Mathematics LibreTexts: 9.2 Arithmetic Sequences and Series
