Arithmetic Sequence Calculator
Find the nth term and sum of a sequence.
Results update as you type.
How it works
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. It is one of the most common patterns in mathematics, underlying everything from simple counting to linear growth. This arithmetic sequence calculator takes the first term, the common difference, and the number of terms, then returns two key values: the nth term, meaning the value of the last term in the range, and the sum of all the terms up to that point. The nth term is found by starting at the first term and adding the common difference once for each step taken, which is n minus one additions to reach the nth position. The sum, known as an arithmetic series, has a neat closed form that avoids adding every term one by one. For a sequence starting at 2 with a common difference of 3, the fifth term is 2 plus four steps of 3, which is 14, and the sum of the first five terms is 40. Students studying sequences and series, and anyone modelling steady, evenly spaced growth, use an arithmetic sequence calculator to find distant terms and totals instantly.
aₙ = a₁ + (n − 1)·d; Sₙ = n/2 · [2a₁ + (n − 1)·d]
An arithmetic sequence adds the same common difference to reach each next term, so the nth term equals the first term plus that difference repeated n minus one times. The sum formula works because pairing the first and last terms, the second and second-to-last, and so on always gives the same total, and there are n halves of such pairs. Multiplying that constant pair sum by n over two totals the whole series.
Source:Standard mathematics
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter first term (a₁), common difference (d) and number of terms (n).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
nth term updates instantly as you type — no button to press.
Frequently asked questions
What is a common difference?
The common difference is the fixed amount added to each term of an arithmetic sequence to get the next one. It is what makes the sequence arithmetic rather than some other pattern. If it is positive the terms increase, if negative they decrease, and if zero every term is the same. You find it by subtracting any term from the one that follows it, and that value stays constant throughout the whole sequence.
How do I find the nth term?
Start with the first term and add the common difference once for every step you take to reach the position you want. Because reaching the nth term takes n minus one steps, the formula is the first term plus n minus one times the common difference. For a sequence beginning at 2 with a difference of 3, the fifth term is 2 plus four threes, which equals 14. The calculator does this directly.
How is the sum of the series calculated?
The sum uses a shortcut discovered by pairing terms from the two ends inward. Each such pair adds to the same total, the first term plus the last, and there are n over two pairs, giving the formula n over two times the sum of the first and last terms. This avoids adding every term individually, so even long sequences are totalled instantly with a single calculation.
What is the difference between a sequence and a series?
A sequence is the ordered list of terms itself, such as 2, 5, 8, 11, and so on. A series is what you get when you add those terms together to form a running total. So the arithmetic sequence lists the values, while the arithmetic series sums them. This calculator reports both: the nth term comes from the sequence, and the sum comes from the corresponding series.
Can the common difference be negative?
Yes. A negative common difference simply means the sequence decreases, with each term smaller than the one before by a fixed amount. For instance, starting at 20 with a common difference of negative four gives 20, 16, 12, 8, and so on. The same formulas apply without change, and the nth term and sum are calculated exactly as with a positive difference. The sum can even turn negative once the terms drop below zero.