🔢 Arithmetic Sequence Calculator: nth Term, Sum and Common Difference

What is the Arithmetic Sequence Calculator?

An arithmetic sequence (also called an arithmetic progression, or AP) is built on one simple idea: you start with a number and keep adding the same value to get the next term. That fixed step is the common difference, written d. If the first term is a (some books call it a1), the sequence runs a, a + d, a + 2d, a + 3d, and so on. When d is positive the sequence grows; when d is negative it shrinks; when d is zero every term is the same. The position of a term is its index, usually written n, starting at 1.

Two formulas do almost all the work, and they describe a sequence in two different ways. The explicit formula (also called the closed form) gives any term directly: the nth term is a + (n - 1)d. It is the fast route, because you can jump straight to the 100th term without listing the first 99. The recursive formula instead defines each term from the one before it: a(n) = a(n - 1) + d, with a(1) = a. Recursion is how a spreadsheet or program usually builds the list, but it is slow for humans because you have to climb one step at a time. Competitors often quote only one form, so it is worth knowing both: the explicit formula answers "what is term n," while the recursive formula answers "how do I get from one term to the next."

The sum of the first n terms, written Sn, is the arithmetic series. There are two equivalent versions. The first, Sn = n/2 x (2a + (n - 1)d), uses only the first term and the difference. The second, Sn = n/2 x (a + L) where L is the last term, multiplies the average of the first and last term by how many terms there are. Both give the identical total, so use whichever inputs you have. This averaging trick is the same one the young Carl Gauss is said to have used to add 1 to 100 in seconds: he paired 1 with 100, 2 with 99, and so on, getting fifty pairs that each sum to 101, for a total of 5050.

A common real task is working backward. If you know two terms but not the difference, use d = (a(m) - a(k)) / (m - k), the change in value divided by the change in position. For example, if the 3rd term is 11 and the 7th term is 27, then d = (27 - 11) / (7 - 3) = 16 / 4 = 4. Once you have d you can recover the first term and use the explicit formula for anything else. If instead you know the first and last term and want to know how many terms there are, rearrange to n = (L - a)/d + 1. That plus one is the step most people forget, and it is why a sequence from 5 to 50 in steps of 5 has ten terms, not nine.

Arithmetic sequences differ from geometric sequences, where you multiply by a fixed ratio instead of adding a fixed difference. A geometric sequence like 2, 6, 18, 54 grows by a factor of 3 each step, while an arithmetic one like 2, 6, 10, 14 grows by adding 4 each step. The quick test: subtract consecutive terms. If those differences are all equal, the sequence is arithmetic and these formulas apply. If instead the ratios between consecutive terms are equal, it is geometric and you need the geometric formulas. Plotting an arithmetic sequence against its index always gives points on a straight line whose slope is d, which is why these sequences model any steady, linear change.

Arithmetic sequences show up far beyond the classroom. Stadium and theater seating often grows by a fixed number of seats per row, so the total capacity is an arithmetic series. Simple-interest savings, straight-line depreciation of an asset, a stack of logs that loses one log per layer, and the page numbers you read at a fixed pace per day are all arithmetic. Knowing the nth term and the sum lets you answer "how much by term n" and "how much in total" without listing everything by hand.

When to use it

• Finding a far-off term, such as the 50th or 100th term, without writing out the whole list by hand.
• Adding up a long run of evenly spaced numbers, like total seats across rows that grow by a fixed amount.
• Working backward to find the common difference or the first term when you only know two terms of the sequence.
• Counting how many terms lie between a known first and last value, using n = (last - first)/d + 1.
• Checking homework answers in algebra, pre-calculus, or AP exams where sequences and series appear.
• Modelling simple linear patterns, such as fixed monthly savings, straight-line depreciation, or a steady weekly production increase.

How to use the Arithmetic Sequence Calculator

1. Enter the first term (a), the value the sequence starts with.
2. Enter the common difference (d), the fixed amount added between consecutive terms (use a negative number for a decreasing sequence).
3. Enter the number of terms (n) you want, as a whole number of 1 or more.
4. Read off the nth term, the sum of those n terms, and the listed sequence below.
5. To find d from two known terms first, subtract them and divide by the gap in their positions, then enter that d here.

Formula & method

Worked examples

nth term and running sum for a = 5, d = 2 (sequence 5, 7, 9, 11, ...)

Quick formula reference for arithmetic sequences

Arithmetic versus geometric sequences at a glance

Common mistakes to avoid

• Using n instead of (n - 1) in the nth term. The nth term is a + (n - 1)d, not a + n x d. The first term takes zero steps from the start, so to reach term n you add d only n - 1 times. Forgetting the minus one shifts every answer by one full step of d.
• Mixing up arithmetic and geometric sequences. Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. If consecutive terms share a ratio rather than a difference, these formulas do not apply, you need the geometric ones instead. Always subtract consecutive terms first to confirm the differences are equal.
• Forgetting that d can be negative or zero. A decreasing sequence has a negative common difference, for example d = -7 gives 100, 93, 86, and so on. Entering it as positive flips the direction and gives the wrong terms and sum. A difference of zero is valid too and produces a constant sequence.
• Counting the number of terms incorrectly. When a sequence is given by its first and last terms, the count is n = (L - a)/d + 1, not just (L - a)/d. The plus one includes the starting term itself, and leaving it off undercounts by one.
• Dividing by the wrong gap when finding d from two terms. The common difference is d = (a(m) - a(k)) / (m - k), so you divide by the difference in positions, not by the difference in values or by the number of terms. For the 3rd and 7th terms, the gap is 7 - 3 = 4, not 7 or 3.
• Treating the sequence and the series as the same thing. The sequence is the list of terms; the series is what you get when you add them up. The nth term is a single number, while the sum Sn totals the first n terms. Reporting one when the question asks for the other is a frequent slip.

Glossary

Arithmetic sequence

A list of numbers in which each term differs from the previous one by a fixed amount, the common difference.

Arithmetic progression (AP)

Another name for an arithmetic sequence, common in many textbooks and exam syllabuses.

Common difference (d)

The constant value added to one term to get the next. It can be positive, negative, or zero.

First term (a or a1)

The number the sequence starts with, the term at position n = 1.

nth term

The term at position n in the sequence, given by the explicit formula a + (n - 1)d.

Explicit formula

A closed-form rule that gives any term directly from its position, a(n) = a + (n - 1)d, without listing earlier terms.

Recursive formula

A rule that defines each term from the one before it, a(n) = a(n - 1) + d with a(1) = a.

Arithmetic series (sum)

The result of adding the terms of an arithmetic sequence. The sum of the first n terms is written Sn.

Frequently asked questions