Arithmetic Sequence Calculator

Written byOleh Soloninupdated 27 September 2026

What you know

First term a₁

Common difference d

Term number n

Result

nth term48
Sum of the first n255
Sum by formulaS₁₀ = (3 + 48) ÷ 2 × 10 = 255
Mean of the terms25.5
Formulaa₁₀ = 3 + (10 − 1) × 5 = 48
Opening terms3, 8, 13, 18, 23
Behaviourincreasing

Each term larger by the same amount

Any term from the first term and the common difference, the sum of the first n terms, and the start of the sequence.

An arithmetic sequence adds the same amount each step. A salary with a fixed annual rise, rows of seats in a hall, days on a calendar — all arithmetic.

Pair the first term with the last, the second with the second-last, and so on: every pair gives the same total. So the sum equals the average of the ends times the count.

How to use

Enter the first term and the common difference.
Enter which term you need.
Read the sum alongside the nth term — it comes from averaging the outer terms.

The sum formula and why it works

With a first term of 3 and a common difference of 5, the tenth term is 3 + 9 × 5 = 48 and the first ten terms sum to 255. The sum is the average of the outer terms times how many there are: (3 + 48) ÷ 2 × 10.

The trick is attributed to Gauss: write the series out twice, the second time backwards, and add in pairs — every pair gives the same total. That also explains why the mean of the series equals the average of its first and last terms, 25.5 here.

First term 3, common difference 5. Worked out with this calculator.
n	nth term	Sum of first n
1	3	3
5	23	65
10	48	255
20	98	1,010

What this calculator shows

The nth term and the sum of the first n terms
The mean of the terms, which equals the average of the two ends
The opening terms and whether the sequence rises or falls

What to keep in mind

Numbering starts at the first term, not a zeroth: a₁ is what you entered.
The common difference may be negative or fractional without changing any formula.

FAQs

How do I find the difference from two terms?

Divide the difference in values by the difference in positions. If a₃ = 13 and a₇ = 33 then d = (33 − 13) ÷ (7 − 3) = 5.

What is the defining property?

Every term except the ends is the average of its two neighbours. It is a quick way to test whether a sequence really is arithmetic.

How do I sum 1 to 100 quickly?

It is a sequence with a₁ = 1, d = 1, n = 100. The sum is (1 + 100) ÷ 2 × 100 = 5050 — the answer the young Gauss is said to have found.

How does it differ from a geometric sequence?

Arithmetic adds the same amount each step; geometric multiplies by the same factor. The first grows linearly, the second exponentially.

What is the tenth term of 3, 8, 13?

48 — the first term plus nine differences, 3 + 9 × 5. The first ten terms add up to 255.

Worked example

The sequence 3, 8, 13, …

Input: a₁ = 3, d = 5, n = 10

Output: a₁₀ = 48, sum 255

Note: Gauss's check: (3 + 48) ÷ 2 × 10 = 255. The mean term is 25.5, exactly the average of the first and last.

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Arithmetic Sequence Calculator

Result

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