Arithmetic Sequence Calculator

First term a₁

Common difference d

Term number n

Result

nth term48
Sum of the first n255
Mean of the terms25.5
Formulaa10 = 3 + (10 − 1) × 5
Opening terms3, 8, 13, 18, 23
Behaviourвозрастающая

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.

The sum formula is credited to Gauss, who as a child added the numbers 1 to 100 instantly by noticing that first-plus-last, second-plus-second-last and so on all give the same pair total. Hence sum equals the average of the ends times the count.

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 young Gauss's answer.

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.

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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