Arithmetic Sequence Calculator
Enter the first term, common difference, and which term number you want to find the nth term and the sum of the first n terms.
How arithmetic sequences work
An arithmetic sequence is a list of numbers where each term increases (or decreases) by the same fixed amount, the common difference (d), from the one before it. The nth term is found as aₙ = a₁ + (n−1)×d, where a₁ is the first term.
The sum of the first n terms is Sₙ = (n/2) × (2a₁ + (n−1)d) — this formula comes from pairing the first and last terms, which always add up to the same total, and multiplying by half the number of terms.
Frequently asked questions
What makes a sequence "arithmetic"?
A sequence is arithmetic if the difference between any term and the one before it is always the same constant value — like 2, 5, 8, 11, ... where each term adds 3.
What if the common difference is negative?
A negative common difference means the sequence decreases — for example, 20, 15, 10, 5, ... has a common difference of −5.
How was the sum formula derived?
Carl Friedrich Gauss famously noticed that pairing the first and last term, the second and second-to-last, and so on, always gives the same sum — multiplying that pair sum by half the number of terms gives the total.