Arithmetic Sequence Calculator
Enter the first term and the common difference, or two known terms. You get the nth term, the sum of the first n terms, the formula and a list of terms.
Worked examples
Term and sum
First term 3, difference 4, n = 10
Term 39, sum 210
3 + 9 × 4 = 39, and 10 ÷ 2 × (3 + 39) = 210.
From two terms
3rd term 11, 7th term 27
First term 3, difference 4
(27 - 11) ÷ 4 = 4.
About Arithmetic Sequence Calculator
An arithmetic sequence goes up or down by the same amount each time. The sequence 3, 7, 11, 15 adds 4 every step, and 4 is the common difference. The nth term is the first term plus (n - 1) times the common difference, so the 10th term here is 3 + 9 × 4, which is 39. The sum of the first n terms is n divided by 2, times the first term plus the last, which for ten terms is 5 × 42, or 210.
If a question gives you two terms rather than the first, the page finds the common difference from the gap between their values divided by the gap between their positions, then works back to the first term. You can also test whether a number is in the sequence: it is if it lands on a whole-numbered position. All the arithmetic uses exact fractions, so a difference of 1/3 stays exact, and the first twelve terms are listed so you can see the pattern for yourself.
How to use Arithmetic Sequence Calculator
- Choose whether you know the first term and difference, or two terms.
- Enter the values. Fractions and negative numbers are fine.
- Enter n, the term you want to find, which also sets the number of terms in the sum.
- Optionally enter a number to test whether it belongs to the sequence.
Common uses
- Finding the nth term for GCSE and A level sequences
- Adding up regular savings or steps
- Seating and stacking patterns with a constant increase
- Checking a pattern from two known terms
Frequently asked questions
What is the formula for the nth term?
aₙ = a₁ + (n - 1)d, where a₁ is the first term and d is the common difference. The page also shows it simplified, such as 4n - 1.
What is the formula for the sum?
Sₙ = n ÷ 2 × (first term + last term). Adding the whole numbers from 1 to 100 gives 100 ÷ 2 × 101 = 5,050.
How do I find the first term from two known terms?
Divide the difference in values by the difference in positions to get d, then take (p - 1) lots of d from the term in position p. The two-terms option does it all.
Can the common difference be negative or a fraction?
Yes. A negative difference gives a falling sequence, and a fractional difference such as 1/2 is kept exact.
How can I tell whether a number is in the sequence?
Work out (value - first term) ÷ d + 1. If that is a whole number of 1 or more, the number is that term of the sequence.