Arithmetic Sequence Calculator (aₙ = a₁ + (n−1)d)

Enter the first term a₁ and common difference d to compute the nth term and partial sums of an arithmetic sequence. Includes a term table and bar chart.

Last updated: 2026-08-25

What is the Arithmetic Sequence Calculator?

The arithmetic sequence calculator automatically works out the nth term (general term) and the sum of the first n terms (partial sum) of an arithmetic sequence — a list of numbers in which the difference between consecutive terms always stays the same. All you need to enter is the first term a₁ and the common difference d. Because the results are shown as both a term table and a bar chart, you can see how the sequence grows or shrinks at a glance, rather than just memorizing the underlying formulas.

Arithmetic sequences are one of the foundational topics in middle school and high school mathematics, built around two formulas: the general term formula aₙ = a₁ + (n−1)d and the partial sum formula Sₙ = n(a₁ + aₙ)/2. This calculator evaluates both instantly, which makes it useful for checking your own hand calculations or comparing several scenarios with different starting values side by side.

Whether you are reviewing homework, preparing for an exam, or simply trying to build intuition for how a sequence with a constant difference evolves term by term, the calculator gives you a complete picture — including every intermediate value — in a matter of seconds.

How to use the Arithmetic Sequence Calculator

Enter the first term a₁ Type in the value of the very first term of the sequence. This becomes the starting point for every other calculation.
Enter the common difference d Type in the constant difference between two consecutive terms. A positive value creates an increasing sequence, and a negative value creates a decreasing one.
Enter n or the number of terms to display Type in either the term number n you want to find, or the number of terms you would like listed in the table.
Check the result, table, and chart The nth term and the sum up to the nth term are displayed instantly, while the table and chart show the overall shape of the sequence.

Tips for getting more out of it

A positive d produces an increasing sequence (e.g. 1, 3, 5, 7, …), a negative d produces a decreasing one, and d = 0 gives a constant sequence.
The nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = n(a₁ + aₙ) / 2.
Type an integer into the "n value" field to instantly compute the nth term and cumulative sum — handy for checking homework answers.
Each bar in the chart corresponds to one term. Equal bar height differences confirm you have a true arithmetic sequence.

Ways to use the Arithmetic Sequence Calculator

Checking homework and exam answers

Instantly compare an nth term or partial sum you calculated by hand against the formula-based result, so you can catch mistakes right away.

Comparing arithmetic and geometric sequences

Placing a sequence with a constant difference side by side with one that has a constant ratio makes the difference in how each one grows much easier to grasp.

Estimating savings plans and even distributions

The logic behind an arithmetic sequence also applies to savings plans that increase by a fixed amount each period, or to distributions that are split with an even, growing gap.

Practicing how to derive the general term

By changing the first term and common difference and watching the results update, you can deepen your understanding of exactly how the general term formula generates a sequence.

Glossary

Arithmetic sequence: A sequence of numbers in which the difference between any two consecutive terms (the common difference) is always the same. Example: 2, 5, 8, 11, … (common difference 3).
Common difference: The difference between two consecutive terms of an arithmetic sequence (the later term minus the earlier term). This value stays constant throughout the entire sequence.
General term: An expression that gives the nth term of a sequence. For an arithmetic sequence it is aₙ = a₁ + (n−1)d.
Partial sum: The total obtained by adding up the terms of a sequence from the first term through the nth term. For an arithmetic sequence it is Sₙ = n(a₁ + aₙ)/2.
Geometric sequence: A sequence of numbers in which the ratio between two consecutive terms (the common ratio) is always the same. It is often contrasted with an arithmetic sequence, where the difference is constant instead.

FAQ

d is the constant difference between consecutive terms (next term − previous term). For example, in 2, 5, 8, 11, … the common difference is d = 3.

Sₙ = n(a₁ + aₙ) / 2, which equals "average of first and last term × number of terms". It can also be written as Sₙ = n(2a₁ + (n−1)d) / 2.

In an arithmetic sequence the difference between consecutive terms is constant. In a geometric sequence the ratio between consecutive terms is constant. For example, 2, 4, 6, 8 is arithmetic (d = 2) while 2, 4, 8, 16 is geometric (r = 2).

Side Note — Gauss and the legend of 1 to 100

The partial-sum formula has a famous origin story. When the mathematician Carl Friedrich Gauss was a schoolboy, his teacher asked the class to add all integers from 1 to 100. While other students laboriously added one by one, Gauss noticed that 1 + 100 = 101, and there are 50 such pairs, giving 5 050 instantly.

That insight — "the first and last terms sum to the same value" — is exactly the formula Sₙ = n(a₁ + aₙ) / 2. Gauss went on to be called the "Prince of Mathematics" and contributed to the normal distribution, prime number theorem, and electromagnetism, but his sharpness was clearly evident from childhood.

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