Arithmetic Sequence Calculator
This arithmetic sequence calculator works out the nth term, the first term, the common difference or the position — fill in any three of the four boxes and the fourth is found with steps, with the terms listed underneath so you can see the answer arrive.
Take 2, 5, 8, 11, … The first term is 2 and 3 is added each time, so the 10th term is 2 + 9 × 3 = 29. Note the 9. Reaching the 10th term does not mean adding 3 ten times, and that single fact costs more marks than anything else in this topic.
What Makes a Sequence Arithmetic
One thing only: the gap between consecutive terms never changes. That fixed gap is the common difference, written d, and you find it by subtracting any term from the one after it.
2, 5, 8, 11 is arithmetic, with d = 3. 7, 4, 1, −2 is arithmetic too, with d = −3 — a negative difference simply means the sequence falls instead of rising. 2, 4, 8, 16 is not arithmetic: the gaps there are 2, 4 and 8, which are not the same number.
That last example is worth a moment, because it is the other big family of patterns. An arithmetic sequence adds a fixed amount each step; something like exponential growth multiplies by a fixed percentage each step. A salary rising by £1,000 a year is arithmetic; a salary rising by 3% a year is not, because the raise itself gets bigger every year. Plotted, the first gives a straight line and the second a curve.
Because the same amount is added over and over, an arithmetic progression — the other name for the same thing, often shortened to AP — is a linear pattern, and that is why its terms sit evenly spaced apart.
The Formula, and Why It Says n − 1
Every arithmetic sequence obeys one rule:
aₙ = a₁ + (n − 1)d
Here a₁ is the first term, d the common difference, n the position you want, and aₙ the term sitting there. Everything on this page comes from rearranging that one line.
The bracket is where marks go missing. It says n − 1, not n, because the first term is already there before any adding starts. Think of a fence: ten posts have nine gaps between them. To walk from the 1st post to the 10th you cross nine gaps, so you add d nine times.
For our example: a₁ = 2, d = 3, n = 10. Nine gaps, so a₁₀ = 2 + 9 × 3 = 29. This calculator prints that gap count as its own line, and then lists the terms — 2, 5, 8, 11, 14, 17, …, 26, 29 — with the one you asked for marked, so you can count it out yourself if you want to.
Finding Any of the Four
The formula has four letters in it, and knowing any three gives the fourth. That is why the calculator has four boxes rather than a fixed set of inputs.
The term at a position. Give a₁, d and n. This is the ordinary case: multiply the gaps by d and add.
The common difference. Give a₁, n and aₙ. The whole rise, shared out over the gaps: d = (aₙ − a₁) ÷ (n − 1). With a₁ = 2, the 3rd term equal to 9 and therefore 2 gaps between them, d comes to 7/2 — and a fractional difference is perfectly ordinary, which is why nothing here is rounded to a decimal.
The position. Give a₁, d and aₙ. Count how many gaps fit between the two terms and add one for the term itself: n = (aₙ − a₁) ÷ d + 1.
The first term. Give d, n and aₙ, and take the added gaps back off: a₁ = aₙ − (n − 1)d.
One case deserves its own answer. Ask where 30 sits in 2, 5, 8, 11, … and there is no honest position to give: 30 would fall 28/3 gaps along, and a term has to sit a whole number of gaps from the start. The sequence steps from 29 straight to 32 and never touches 30, and the calculator says exactly that rather than rounding to something that looks like an answer.
How to Use This Arithmetic Sequence Calculator
The screen has three parts: the answer band at the top, four boxes below it, and the keypad underneath.
1. Fill in any three boxes. The one you leave empty turns dashed with a ? in it, so you can see at a glance which one is about to be worked out. Whole numbers, decimals such as 0.25 and fractions such as 3/4 are all accepted.
2. Press =. The answer appears at the top, and the empty box turns green and shows it too, with ↑ shown above beside its label so there is no doubt which box the answer belongs to.
3. Open the working. Show the working gives the formula, the gap count written out as n − 1, the rearrangement for whichever box was empty, the terms themselves, and finally the same journey walked one gap at a time as a check.
4. Or fill in all four. If you already have an answer and want it checked, type all four numbers. The calculator works the first three through and compares, then tells you whether they agree — and if they do not, what the fourth number should have been.
Every key and control, in one place:
| first term a₁ | The number the sequence starts at |
| common difference d | What is added each step; negative for a falling sequence |
| position n | Which term you want, counting the first as 1. A whole number |
| term aₙ | The value sitting at that position |
| ← → ↑ ↓ | Move between the four boxes |
| ± | Flips the sign of the box you are in |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way through |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ | Deletes one character from the box you are in |
| AC | Clears all four boxes |
| = | Works it out |
| My keyboard | Raises your own keyboard instead of the pad, for that box |
| copy | The icon in the band copies the answer |
You can also type straight into a box with a keyboard — the pad is there for phones, not instead of typing.
Arithmetic Sequence FAQ
What is an arithmetic sequence?
What is the formula for the nth term?
Why is it n minus 1 and not n?
How do you find the common difference?
Can the common difference be negative or a fraction?
How do you know if a number belongs to a sequence?
One formula, four letters, and a bracket that says n − 1 for a reason. Get the gap count right and the rest is arithmetic you could do on paper — which is exactly why this page shows you the terms as well as the answer.