Enter the first term, the common difference and how many terms you want to get the last term and the total sum of an arithmetic sequence — a list where each term increases (or decreases) by the same fixed amount. The leading terms are listed so you can check the pattern at a glance.
Enter a first term, a common difference and how many terms.
How it works
An arithmetic sequence adds a constant step, the common difference d, to move from each term to the next: starting at a₁, the terms are a₁, a₁+d, a₁+2d, and so on. The nth term is therefore aₙ = a₁ + (n−1)·d — you add the difference (n−1) times, not n times, because the first term is already there.
The sum of the first n terms is Sₙ = n ÷ 2 × (a₁ + aₙ): pair the first and last terms, then the second and second-to-last, and every pair adds up to the same total. The difference can be negative (a decreasing sequence) or zero (every term the same). The number of terms must be a whole number of at least 1; decimals are accepted for the first term and the difference.
Practical examples
Seats in a theatre
A row has 20 seats and each row behind adds 2 more. Row 15 has a₁₅ = 20 + 14×2 = 48 seats, and the first 15 rows hold 15 ÷ 2 × (20 + 48) = 510 seats in total.
Gauss’s schoolroom sum
Adding the whole numbers 1 through 100 is an arithmetic sequence with a₁ = 1, d = 1, n = 100. The sum is 100 ÷ 2 × (1 + 100) = 5,050 — the trick a young Gauss is said to have spotted.
A countdown
Starting at 100 and decreasing by 7 each step, the sequence is 100, 93, 86, … With d = −7, the 10th term is 100 + 9×(−7) = 37.
Frequently asked questions
What is an arithmetic sequence?
A list of numbers where the gap between consecutive terms is always the same. That constant gap is the common difference. 3, 7, 11, 15 is arithmetic with a common difference of 4.
What is the common difference?
It is the fixed amount added to each term to get the next one, written d. You find it by subtracting any term from the one after it: in 5, 8, 11 the difference is 8 − 5 = 3.
What is the formula for the nth term?
aₙ = a₁ + (n−1)·d, where a₁ is the first term, d is the common difference and n is the position. The (n−1) is because you take (n−1) steps to reach the nth term from the first.
How is the sum calculated?
Sₙ = n ÷ 2 × (a₁ + aₙ), the number of terms times the average of the first and last term. It works because terms pair up symmetrically to the same total. Equivalently, Sₙ = n ÷ 2 × (2a₁ + (n−1)d).
Can the common difference be negative?
Yes. A negative difference makes a decreasing sequence, like 20, 17, 14, 11 with d = −3. The formulas are identical; the terms simply go down.
What if the difference is zero?
Then every term equals the first term, and the sum is just n times that value. It is a valid, if trivial, arithmetic sequence.
How is an arithmetic sequence different from a geometric one?
An arithmetic sequence adds a fixed amount each step; a geometric sequence multiplies by a fixed ratio. 2, 5, 8, 11 is arithmetic (add 3); 2, 6, 18, 54 is geometric (multiply by 3).
Why do you only show the first several terms?
The preview lists up to twelve terms so you can confirm the pattern. The nth term and the sum are still computed for the full length you enter, even if that is thousands of terms.
Do the first term and difference have to be whole numbers?
No. They can be decimals — for example a first term of 1.5 and a difference of 0.5. Only the number of terms must be a whole number, since you can’t have a fraction of a term.
Is my input sent anywhere?
No. The sequence is generated and summed in your browser. Nothing you type is uploaded or logged.
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