Arithmetic Sequence Calculator
Find the nth term and the sum of an arithmetic sequence from its first term and common difference.
How it works
nth term = a + (n − 1)d · sum of n terms = n ÷ 2 × (2a + (n − 1)d)
An arithmetic sequence adds the same fixed amount, the common difference d, to move from one term to the next. The nth term jumps straight to any position without listing the ones before it, and the sum formula works because pairing the first and last terms, the second and second-last, and so on, always gives the same total — so the sum is the average of the endpoints times how many terms there are.
Worked example
With first term 3 and common difference 5, the terms run 3, 8, 13, 18 … The 10th term is 3 + (10 − 1) × 5 = 48. Their sum is 10 ÷ 2 × (2 × 3 + 9 × 5) = 5 × 51 = 255.
Sources: Arithmetic progression — Wikipedia
Frequently asked questions
What makes a sequence arithmetic?
The difference between consecutive terms is constant. 3, 8, 13, 18 is arithmetic (always +5); 3, 6, 12, 24 is not, because that pattern multiplies instead.
Why does the sum formula use the average of the endpoints?
Pairing terms from the outside in — first with last, second with second-last — each pair sums to the same value. There are n terms, so the total is n times their average, which is (first + last) ÷ 2.
Can the common difference be negative?
Yes. A negative d makes a decreasing sequence, like 20, 17, 14, 11. The same nth-term and sum formulas apply without any change.
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