Arithmetic Sequence Calculator

Background.

An arithmetic sequence is the simplest interesting sequence in mathematics: a list of numbers where you get from each term to the next by adding the same fixed amount every single time. That fixed amount is the common difference, written d. Start at 3 and keep adding 5 and you get 3, 8, 13, 18, 23 — an arithmetic sequence with a₁ = 3 and d = 5. This calculator handles that sequence in every direction. Give it the first term, the common difference and a position, and it returns the term sitting at that position. Give it two terms and a position count and it works backwards to the common difference. Give it a target value and it tells you which position that value occupies — or refuses, correctly, if the value is not actually a term of the sequence at all.

Four things fully describe an arithmetic sequence: the first term a₁, the common difference d, how many terms you are looking at (n), and the value of the last of those terms (aₙ). Knowing any three of the four pins down the fourth, which is exactly what the mode selector at the top of the calculator does. Whichever one you solve for, all four are reported together, along with the sum of the whole run and the explicit rule you can write down in a homework answer. There is no second page to visit and no re-entering of numbers to get the sum after you have found the term.

The page is 1-indexed throughout, and that convention is worth stating before you read a single result. The first term is a₁, not a₀, so the n-th term is a₁ + (n − 1)d and n is always a positive whole number. The subtraction of 1 in that formula is the single most common source of off-by-one errors in sequence homework: you take d steps to get from term 1 to term 2, so getting to term 12 takes eleven steps, not twelve. If your textbook indexes from zero — a few discrete-mathematics and computer-science texts do — every index on this page will sit one higher than the one in your book.

The partial sum matters as much as the term itself, and it has an elegant history. The story that a young Carl Friedrich Gauss added the whole numbers from 1 to 100 in seconds by pairing 1 with 100, 2 with 99, and so on to get fifty pairs of 101 is the standard classroom illustration of the sum formula. Every pair drawn inward from the two ends of an arithmetic sequence adds to the same total, a₁ + aₙ, and there are n/2 such pairs — which is precisely Sₙ = n(a₁ + aₙ)/2. Enter a₁ = 1, d = 1, n = 100 into this page and you get the 5050 that story turns on.

The distinction worth drawing before you use the result: an arithmetic sequence is a straight line restricted to whole-number positions. Its terms plot as evenly spaced points on a line whose gradient is d and whose value at n = 1 is a₁. That is why the linear-equation and slope calculators feel adjacent — but they work over the continuous real numbers, where there is no first term, no term count and no partial sum. If you have a set of observed numbers and do not yet know whether they form an arithmetic sequence at all, the sequence calculator identifies the family for you first. If your terms multiply rather than add, you want the geometric sequence calculator instead. And if you need a sum whose index runs from something other than 1, or a sum of squares or cubes, the sum of series calculator handles the sigma notation directly.

What is arithmetic sequence calculator?

An arithmetic sequence — also called an arithmetic progression — is a sequence of numbers in which the difference between any two consecutive terms is constant. OpenStax's College Algebra 2e puts it as: 'The difference between any two consecutive terms is a constant. This constant is called the common difference.' That constant is written d, and it can be positive (an increasing sequence), negative (a decreasing sequence) or zero (a constant sequence where every term is the same). The n-th term has the explicit formula aₙ = a₁ + d(n − 1), and the sum of the first n terms — called an arithmetic series — is Sₙ = n(a₁ + aₙ)/2. The Encyclopedia of Mathematics records the same two formulas and adds a neat characteristic test: in an arithmetic progression every term is the average of its two neighbours, aₙ = (aₙ₊₁ + aₙ₋₁)/2. Arithmetic sequences appear anywhere a quantity changes by a fixed amount per step: seats added row by row in an auditorium, a fixed monthly repayment schedule ignoring interest, degree markings on a scale, or the positions of evenly spaced fence posts.

How to use this calculator.

1. 01Pick what you want to find from the mode selector — the nth term, the common difference, the first term, or the number of terms.
2. 02Fill in the three fields that mode needs. The field it solves for is ignored, so you can leave whatever is already in it.
3. 03Read the hero number, then the sum of the run, the explicit rule and the sequence preview beneath it.
4. 04Check the preview line: it tells you the opening terms, the final term, the count, and whether the sequence is increasing, decreasing or constant.
5. 05If you are solving for the number of terms and get an error, the value you entered for aₙ is not on this sequence — the message tells you the fractional position it would occupy, which is usually enough to spot the typo.

The formula.

Getting from term 1 to term n takes n − 1 steps, and each step adds d, so aₙ = a₁ + (n − 1)d. Rearranging that single equation gives the three inverse solves this page offers: d = (aₙ − a₁)/(n − 1) needs at least two terms, because a single term has no gap to measure; a₁ = aₙ − (n − 1)d; and n = (aₙ − a₁)/d + 1, which requires d ≠ 0 and only makes sense when the quotient lands on a whole number. The sum comes from the pairing argument: write the run forwards and backwards under itself, add the two rows column by column, and every column totals a₁ + aₙ. There are n columns, and you have counted each term twice, so Sₙ = n(a₁ + aₙ)/2. Substituting the term formula gives the equivalent expansion Sₙ = n[2a₁ + (n − 1)d]/2, and the test suite asserts the two forms agree across a fixed sweep of parameters. On rounding: nothing is rounded at an intermediate step. Every quantity is carried at full decimal precision and rounded once, at the boundary, to 10 decimal places, which is why entering a₁ = 0.1 with d = 0.2 gives a third term of exactly 0.5 rather than the 0.5000000000000001 a floating-point implementation returns. The one exactness test — 'is aₙ really a term of this sequence' — is applied to the unrounded quotient before any rounding happens, so a near-miss is rejected rather than quietly snapped to the nearest term. The increasing/decreasing/constant wording in the preview is likewise classified from the unrounded common difference, not from the rounded value shown in the results panel.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax (Rice University), College Algebra 2e, §9.2 'Arithmetic Sequences' — defines the common difference ('The difference between any two consecutive terms is a constant. This constant is called the common difference.') and gives the explicit formula aₙ = a₁ + d(n − 1). Retrieved 2026-07-29; open access.
2. [2]OpenStax (Rice University), College Algebra 2e, §9.4 'Series and Their Notations' — gives the arithmetic series formula Sₙ = n(a₁ + aₙ)/2. Retrieved 2026-07-29; open access.
3. [3]Encyclopedia of Mathematics (European Mathematical Society), 'Arithmetic progression' — independent check on both formulas: general term aₙ = a + (n − 1)d, sum Sₙ = (a₁ + aₙ)n/2, plus the characteristic property aₙ = (aₙ₊₁ + aₙ₋₁)/2. Agrees with OpenStax on the worked example (S₁₂ = 366). Retrieved 2026-07-29; open access.
4. [4]Wolfram MathWorld, 'Arithmetic Series' — the pairing argument that produces Sₙ = ½n(a₁ + aₙ), and the Gauss anecdote it is usually taught with. Retrieved 2026-07-29; open access.
5. [5]OpenStax (Rice University), Calculus Volume 1, §5.1 'Approximating Areas', rule 'Sums and Powers of Integers' — Σᵢ₌₁ⁿ i = n(n + 1)/2, the a₁ = 1, d = 1 special case used as the Gauss anchor test in this calculator's suite. Retrieved 2026-07-29; open access.