Arithmetic Sequence Calculator: nth Term and Sum
List the terms of an arithmetic sequence from its first term and common difference, with the nth-term formula, the last term and the exact sum. Fractions and decimals stay exact. Nothing is uploaded.
- Formula for aₙ
- aₙ = a₁ + (n - 1)d = n
- Term a₁₀
- 10
- Formula for Sₙ
- Sₙ = n(a₁ + aₙ) / 2
- Sum S₁₀
- 55
About the arithmetic sequence calculator
This page lists the terms of an arithmetic sequence and works out its nth term and sum. Enter the first term a₁, the common difference d and the number of terms n, and the terms appear at once, together with the formula for the nth term, the value of the last term aₙ and the sum Sₙ. The first term and the difference may be whole numbers, decimals such as 0.25 or fractions such as 1/3, and the arithmetic is exact: no term is ever rounded, so a difference of 0.1 gives 0.1, 0.2, 0.3 rather than 0.30000000000000004. Everything is calculated in your browser, and nothing is uploaded.
How to use it
Steps
- Type the First term a₁, for example 3.
- Type the Common difference d, for example 2. A negative difference makes the sequence fall.
- Set the Number of terms, up to 10,000.
- Read the formula, the last term and the sum, then copy or download the list with a comma, space or new-line separator.
The formulas
nth term
Each term is the previous one plus d, so the nth term is
The page also writes it in simplified form: with a₁ = 3 and d = 2 it reads aₙ = 2n + 1, so a₁₀₀ = 201. A difference of 0 gives a constant sequence.
Sum of the first n terms
Pair the first term with the last, the second with the second-to-last, and so on; every pair has the same total. That gives
For 3, 5, 7, 9, 11 the sum is 5 × (3 + 11) / 2 = 35. The sum of 1 to 100 is 5050, the result attributed to the young Gauss.
Worked examples
- Seats in a theatre: 20 seats in the first row and 2 more in each row give 20, 22, 24, …; with 25 rows the last row has 68 seats and the theatre has 1100.
- Savings: putting aside 50 in the first month and 10 more each month gives 50, 60, 70, …; after 12 months you have saved 1260.
- Fractions: a first term of 0 and a difference of 1/3 give 0, 1/3, 2/3, 1, shown as exact fractions because their decimals repeat.
Privacy
The terms and sums are worked out by code running on this page, so the values you type are never sent anywhere and are forgotten when you close the tab.
Related tools
For a sequence that multiplies by a fixed ratio instead, use the Geometric Sequence Calculator. The full Number Sequence Generator adds even and odd numbers, powers, squares and cubes, Fibonacci-like series, linear recurrences and more, and the Number List Calculator finds the mean, median and other statistics of a list you paste.
Open-source credits
The arithmetic is implemented in-house in TypeScript with the built-in BigInt type and exact fractions in lowest terms; no third-party library is used. The even and odd numbers derived from it are tested against OEIS A005843 and A005408.