Arithmetic Sequence Calculator

Find the nth term and the sum of an arithmetic sequence from the first term, the common difference and how many terms.

nth term (aₙ)
Sum (Sₙ)

How to Work With an Arithmetic Sequence

Enter the first term, common difference and term count.

1

Enter the sequence

a₁ is the first term, d is the constant amount added each step, and n is how many terms.

2

Read the nth term

aₙ = a₁ + (n − 1)d gives the value of the nth term.

3

Read the sum

Sₙ = n(2a₁ + (n − 1)d) ⁄ 2 adds the first n terms.

Frequently Asked Questions

What is an arithmetic sequence?
An arithmetic sequence is a list of numbers in which each term differs from the previous one by a fixed amount, called the common difference d. In 3, 7, 11, 15 the common difference is d = 4, because every step adds 4. A negative d makes the sequence decrease, as in 20, 17, 14, 11 with d = −3. Only the first term and the common difference are needed to generate every later term.
What is the formula for the nth term?
The nth term of an arithmetic sequence is aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference. For 3, 7, 11, … with a₁ = 3 and d = 4, the 10th term is 3 + 9 × 4 = 39. The (n − 1) matters: the first term has taken no steps, so a₁ = a₁ + 0 × d. Rearranging also lets you find the position of a known term, n = (aₙ − a₁) ⁄ d + 1.
How do you find the sum?
The sum of the first n terms is Sₙ = n(2a₁ + (n − 1)d) ⁄ 2, which is n times the average of the first and last terms. For 3, 7, 11, … the first 10 terms sum to 10(2 × 3 + 9 × 4) ⁄ 2 = 10 × 42 ⁄ 2 = 210. The equivalent form Sₙ = n(a₁ + aₙ) ⁄ 2 is quicker when you already know the last term: 10(3 + 39) ⁄ 2 = 210. A partial sum like this is called an arithmetic series.
Is my data sent anywhere?
Nothing you enter into the arithmetic sequence calculator leaves your device. The arithmetic sequence calculator does all of its arithmetic client-side in JavaScript, so the first term, common difference and term count you enter are never uploaded to a server. No account is needed and no calculation is stored or logged. Once the page has loaded it works with no network connection at all.